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Using a Graphic Calculator to Promote a Preservice Teacher's Didactic Knowledge of Functions

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Title: Using a Graphic Calculator to Promote a Preservice Teacher's Didactic Knowledge of Functions
Language: English
Authors: Floriano Viseu, Ana Paula Aires, Sara Cruz
Source: International Journal of Mathematical Education in Science and Technology. 2025 56(8):1457-1475.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 19
Publication Date: 2025
Document Type: Journal Articles
Reports - Research
Tests/Questionnaires
Education Level: Higher Education
Postsecondary Education
Secondary Education
Grade 11
High Schools
Descriptors: Mathematics Instruction, Preservice Teachers, Graphing Calculators, Secondary School Teachers, Preservice Teacher Education, Grade 11, Knowledge Level, Pedagogical Content Knowledge, Foreign Countries, Lesson Plans, Planning
Geographic Terms: Portugal
DOI: 10.1080/0020739X.2025.2492779
ISSN: 0020-739X
1464-5211
Abstract: The initial teacher education courses for Mathematics aim to promote an adequate education in mathematics and mathematics didactics. Equally or even more important than this education is understanding how the preservice teacher should build new knowledge from the activities they undertake. Considering the teaching practicum, this study aims to understand the contribution of the graphic calculator that complies with the guidelines set by the Portuguese Ministry of Education for secondary education in promoting the didactic knowledge of a preservice teacher in the teaching of functions in 11th grade within the Portuguese education system. Adopting a qualitative and interpretive methodology, following a design-based approach, data were collected through her lesson plans, reflections and practicum report. Regarding content knowledge of functions, the preservice teacher identifies different representations and reveals knowledge of facts and procedures. In terms of specialised content knowledge, she explores the graphic calculator to connect representations and identifies different task-solving strategies. About content knowledge and the curriculum, she knows the curricular recommendations about the teaching of functions and the integration of the graphic calculator in mathematics lessons. Regarding content knowledge and students, she identifies factors that enhance or inhibit understanding and evaluates students' comprehension of the topics studied.
Abstractor: As Provided
Entry Date: 2025
Accession Number: EJ1481510
Database: ERIC
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  Value: <anid>AN0187409024;imt01aug.25;2025Aug21.03:39;v2.2.500</anid> <title id="AN0187409024-1">Using a graphic calculator to promote a preservice teacher's didactic knowledge of functions </title> <p>The initial teacher education courses for Mathematics aim to promote an adequate education in mathematics and mathematics didactics. Equally or even more important than this education is understanding how the preservice teacher should build new knowledge from the activities they undertake. Considering the teaching practicum, this study aims to understand the contribution of the graphic calculator that complies with the guidelines set by the Portuguese Ministry of Education for secondary education in promoting the didactic knowledge of a preservice teacher in the teaching of functions in 11th grade within the Portuguese education system. Adopting a qualitative and interpretive methodology, following a design-based approach, data were collected through her lesson plans, reflections and practicum report. Regarding content knowledge of functions, the preservice teacher identifies different representations and reveals knowledge of facts and procedures. In terms of specialised content knowledge, she explores the graphic calculator to connect representations and identifies different task-solving strategies. About content knowledge and the curriculum, she knows the curricular recommendations about the teaching of functions and the integration of the graphic calculator in mathematics lessons. Regarding content knowledge and students, she identifies factors that enhance or inhibit understanding and evaluates students' comprehension of the topics studied.</p> <p>Keywords: Initial teacher education; didactic knowledge; mathematics; graphic calculator</p> <hd id="AN0187409024-2">1. Introduction</hd> <p>Didactic knowledge is a multifaceted form of knowledge that teachers develop throughout their professional activity to make mathematical content understandable to their students (Ponte & Chapman, [<reflink idref="bib29" id="ref1">29</reflink>]). Its origins lie, from a formal perspective, in academic studies, but it is in the classroom practice that it is consolidated and shapes the teacher as an expert as they accumulate teaching experience (Copur-Gencturk & Tolar, [<reflink idref="bib11" id="ref2">11</reflink>]; Escudero-Ávila & Carrillo Yáñez, [<reflink idref="bib15" id="ref3">15</reflink>]; Schön, [<reflink idref="bib32" id="ref4">32</reflink>]). Initial teacher education courses play a crucial role in this development by providing studies on topics incorporating didactic knowledge. This suitability stems from the decisions made in the facilitation of teaching and learning activities, which have pedagogical implications, such as how the future teacher values what the student says and does (Ponte, [<reflink idref="bib28" id="ref5">28</reflink>]). Equally or even more important than this education is knowing how the future teacher should construct new knowledge from their activities (Ebby, [<reflink idref="bib12" id="ref6">12</reflink>]; McDuffie, [<reflink idref="bib22" id="ref7">22</reflink>]), with reference to their conceptions and knowledge. Ponte and Chapman ([<reflink idref="bib29" id="ref8">29</reflink>]) advocate the integration of content, pedagogy, and reflection in the construction of own knowledge. This integration can result from experiences that take place simultaneously in both school and university, such as during the 'Professional Internship' year.</p> <p>Teaching practice and reflection on that practice constitute a fundamental pairing in teacher education (Ponte & Chapman, [<reflink idref="bib29" id="ref9">29</reflink>]; Viseu & Ponte, [<reflink idref="bib37" id="ref10">37</reflink>]). The activity that results from this interaction is particularly important in initial teacher education, as it usually represents the first opportunity for preservice teachers to confront the reality of the classroom with their personal theories, both those formed through experiential processes (as students) and through structured education processes during their teacher education course. These formative processes, which include knowledge from various disciplinary areas (particularly Mathematics and its Didactics) developed throughout the course, as well as the knowledge gained from professional practice during the teaching practicum, interrelate to form the teacher's didactic knowledge, with emphasis on the knowledge of mathematical content and the knowledge of teaching practice. These two types of knowledge reflect the teacher's understanding of the curriculum and their knowledge of how students learn (Appova & Taylor, [<reflink idref="bib1" id="ref11">1</reflink>]; Ponte, [<reflink idref="bib27" id="ref12">27</reflink>]). The integration of the graphical calculator into teaching and learning strategies serves as an example of didactic approaches that embody these characteristics, incorporating a strong experimental component (Martins & Domingos, [<reflink idref="bib20" id="ref13">20</reflink>]; Viseu & Menezes, [<reflink idref="bib36" id="ref14">36</reflink>]). The relevance of the graphing calculator in this study is due to the recommendations of official curriculum documents (Ministério da Educação, [<reflink idref="bib24" id="ref15">24</reflink>]), which translate into the mandatory use of this resource in secondary school[<reflink idref="bib1" id="ref16">1</reflink>] mathematics lessons. Such recommendations mean that teachers in general, and preservice teachers in particular, should explore and integrate it into their teaching practice in order to make mathematical knowledge meaningful (Chen & Lai, [<reflink idref="bib7" id="ref17">7</reflink>]; Clark-Wilson et al., [<reflink idref="bib8" id="ref18">8</reflink>]). The study presented here arises at the intersection of these two areas: the initial education of mathematics teachers and the graphical calculator as a mediating resource for learning mathematics. In particular, this study aims to understand the contribution of graphic calculator that comply with the guidelines set by the Portuguese Ministry of Education for secondary education in promoting the didactic knowledge of a preservice teacher in the teaching of functions to 11th grade, within the Portuguese education system, in his teaching internship.</p> <hd id="AN0187409024-3">2. The concept and the nature of didactic knowledge</hd> <p>Research has placed special emphasis on the knowledge that teachers need to teach and on how this knowledge is acquired (Ball et al., [<reflink idref="bib3" id="ref19">3</reflink>]; Ponte & Chapman, [<reflink idref="bib29" id="ref20">29</reflink>]). The relevance of this knowledge emerged from Shulman's ([<reflink idref="bib33" id="ref21">33</reflink>]) critical stance regarding the tendency of research to focus more on pedagogical aspects than on teachers' own knowledge. In recognising the existence of specific knowledge required for teaching, the author organises this knowledge into various dimensions, giving particular emphasis to 'pedagogical content knowledge' as it highlights the teacher's ability to enable students to understand mathematical content. The importance of this knowledge lies in the connection between content knowledge and teaching practice, reflecting the value placed on both the discussion of content when developing teaching strategies and the discussion of teaching strategies with regard to the content (Ball et al., [<reflink idref="bib3" id="ref22">3</reflink>]).</p> <p>Despite the recognition of Shulman's work in characterising teachers' professional knowledge, some authors problematise his notion of pedagogical content knowledge, such as Ponte and Chapman ([<reflink idref="bib29" id="ref23">29</reflink>]) and Ball et al. ([<reflink idref="bib3" id="ref24">3</reflink>]). According to Ponte and Chapman ([<reflink idref="bib29" id="ref25">29</reflink>]), Shulman places little emphasis on teachers' actions and does not include students or the curriculum in this knowledge, which results in the notion of pedagogical content knowledge highlighting a more declarative conception of teachers' knowledge than one oriented towards or embedded in action. Ball et al. ([<reflink idref="bib3" id="ref26">3</reflink>]) argue that the knowledge required for teachers to teach incorporates other elements beyond content knowledge and pedagogical knowledge, commonly referred to as 'didactic knowledge'. According to Ponte ([<reflink idref="bib27" id="ref27">27</reflink>]), this knowledge encompasses four dimensions: (i) knowledge of mathematics, including its internal interrelations and with other subjects, as well as its forms of reasoning, argumentation, and validation; (ii) knowledge of the curriculum, including its major purposes and objectives, and its vertical and horizontal articulation; (iii) knowledge of the student and their learning processes; and (iv) knowledge of classroom processes.</p> <p>Building on Shulman's notion of pedagogical content knowledge, Ball et al. ([<reflink idref="bib3" id="ref28">3</reflink>]) developed the concept of 'mathematical knowledge for teaching', which is organised around two dimensions, each comprising three categories: (<reflink idref="bib1" id="ref29">1</reflink>) content knowledge, which includes common content knowledge (knowledge possessed by anyone with a mathematical background), horizon content knowledge (knowledge of how mathematical content is related across the curriculum at different educational levels), and specialised content knowledge (knowledge that enables the Mathematics teacher to explain to students the reasoning behind mathematical facts and procedures); (<reflink idref="bib2" id="ref30">2</reflink>) pedagogical content knowledge, with a focus on teaching, which is subdivided into knowledge of content and students, content and teaching, and content and curriculum.</p> <p>Regardless of the model considered, a mathematics teacher's didactic knowledge is an amalgamation of essential knowledge required to teach this subject's content. This knowledge becomes explicit both during the planning and delivery of lessons, as well as in the reflection and evaluation of teaching and student learning.</p> <p>The teacher's didactic knowledge is built through multifaceted forms that result from formal and informal interactions in education settings or with peers (Boero et al., [<reflink idref="bib5" id="ref31">5</reflink>]). Cochran-Smith and Lytle ([<reflink idref="bib10" id="ref32">10</reflink>]) distinguishes two types of didactic knowledge: formal and practical. Formal knowledge reflects academic knowledge, which, according to Schön ([<reflink idref="bib31" id="ref33">31</reflink>]), is acquired in terms of facts, rules, and procedures in its application to situations encountered by the teacher in practice contexts. However, certain unforeseen and conflicting classroom situations mean that such knowledge is not always sufficient to resolve them. According to Schön ([<reflink idref="bib31" id="ref34">31</reflink>]), the problem situations of practice are not always well-defined and do not always have a clear solution. Such situations require teachers to improvise and test situational strategies (Schön, [<reflink idref="bib31" id="ref35">31</reflink>]). In having to interpret, make decisions, and resolve particular situations, the teacher develops a type of practical knowledge (Elbaz, [<reflink idref="bib13" id="ref36">13</reflink>]), which results from experience and the reflection upon it (Schön, [<reflink idref="bib31" id="ref37">31</reflink>]). This is a type of knowledge that is neither objective nor pre-existing (Campbell, [<reflink idref="bib6" id="ref38">6</reflink>]), but rather personal (Elbaz, [<reflink idref="bib13" id="ref39">13</reflink>]) and tacit (Schön, [<reflink idref="bib31" id="ref40">31</reflink>]). Practical knowledge is built through action and is legitimised the more it is combined with theory (Ponte & Chapman, [<reflink idref="bib29" id="ref41">29</reflink>]), which serves to analyse personal knowledge and its usefulness (Elbaz, [<reflink idref="bib13" id="ref42">13</reflink>]). It represents the theorisation of practice in relation to teaching situations, based on reflection in and on action (Schön, [<reflink idref="bib32" id="ref43">32</reflink>]). Reflection establishes connections between formal knowledge and practical knowledge, allowing the production and reconstruction of practical knowledge related to teaching activities. It is the relationship between theoretical knowledge and practice, alongside the ability to analyse that practice, that promotes the teacher's didactic knowledge (Ponte, [<reflink idref="bib27" id="ref44">27</reflink>]).</p> <p>Research in mathematics education identifies several strategies to develop teachers' didactic knowledge, particularly for preservice teachers. Although practical contexts are ideal for this development, Ponte and Chapman ([<reflink idref="bib29" id="ref45">29</reflink>]) argue that teacher education courses can address aspects such as, such as lesson planning, learning theories, the exploration of teaching materials, and students' cognitive development.</p> <p>Engaging future teachers with classroom experiences during their initial education helps them understand teaching phenomena (Viseu & Menezes, [<reflink idref="bib36" id="ref46">36</reflink>]). Regarding classroom experiences, the teaching practicum year is an ideal setting for future teachers to enhance their didactic knowledge through the development and implementation of lesson plans and reflecting on their actions (Viseu & Menezes, [<reflink idref="bib36" id="ref47">36</reflink>]). This experience connects content with teaching methods, influencing decisions about: content; the selection of tasks; the time dedicated to completing tasks; the organisation and management of activities; the identification of possible student difficulties; and the use of teaching materials.</p> <p>Given the abstract nature of the mathematics concepts, teaching materials can become highly effective allies in enhancing mathematics classroom activities, such as the graphical calculator (Viseu et al., [<reflink idref="bib38" id="ref48">38</reflink>]). The graphical calculator is a resource that greatly contributes to the development of the didactic knowledge of future mathematics teachers (Viseu & Menezes, [<reflink idref="bib36" id="ref49">36</reflink>]), regarding knowledge of teaching, students, specialised content, and the curriculum.</p> <p>The effective use of the graphical calculator enables teachers to develop a deeper understanding of how to articulate mathematical concepts and how to improve their pedagogical practice. The integration of graphical calculators into pedagogical practice enhances the future teacher's knowledge of their students by fostering collaborative learning (Balantes & Tonga, [<reflink idref="bib2" id="ref50">2</reflink>]), identifying students' difficulties in understanding mathematical concepts and monitoring students' progress (Tabach & Trgalová, [<reflink idref="bib34" id="ref51">34</reflink>]). Kastberg and Leatham ([<reflink idref="bib18" id="ref52">18</reflink>]) consider that the teacher's specialised knowledge of using the graphical calculator is closely linked to their ability to assess and respond to students' difficulties. These authors argue that teachers who are proficient in the use of the graphical calculator are more effective in adapting their teaching strategies to support students in solving complex problems.</p> <p>In formative terms, the exploration of the graphical calculator in promoting the didactic knowledge of the future teacher is aligned with the mathematical meaning derived from this exploration, referred to by Rabardel ([<reflink idref="bib30" id="ref53">30</reflink>]) and Trouche ([<reflink idref="bib35" id="ref54">35</reflink>]) as instrumental genesis, which results from the symbiosis between usage schemes, management of the specific characteristics and properties of the calculator, and instrumented action schemes, which translate the actions performed on the calculator into meaningful representations of the concepts being studied.</p> <hd id="AN0187409024-4">3. Methodology</hd> <p>This study aims to understand the contribution of graphic calculator that comply with the guidelines set by the Portuguese Ministry of Education for secondary education in promoting the didactic knowledge of a preservice teacher in the teaching of functions to 11th grade within the Portuguese education system, within the context of her teaching practicum. The selection of this preservice teacher was the result of a search we carried out in the library repository of Portuguese universities that offer initial teacher training courses in mathematics. In this search, we looked for internship reports whose title included the graphing calculator in the teaching or learning of functions. The reason for this mathematical theme was the compulsory use of this resource in secondary school math classes. After identifying the most recent report, we contacted the future teacher with the aim of providing us with other documents that included elements of her teaching practice. In response to our request, she sent us the portfolio she prepared during the year of her practicum internship. She is a 22-year-old preservice teacher with no prior teaching experience. Her relationship with the graphical calculator was limited to its use during her secondary school years for solving tasks that were presented to her. Given the nature of the objective of this study, we adopted a qualitative and interpretive approach to understand the meanings that the preservice teacher assigns to her actions in the development of her pedagogical practice (Erickson, [<reflink idref="bib14" id="ref55">14</reflink>]; McMillan & Schumacher, [<reflink idref="bib23" id="ref56">23</reflink>]), following a design-based research approach with three cycles (Cobb et al., [<reflink idref="bib9" id="ref57">9</reflink>]).</p> <p>Design-based research is a research method that involves the design and development of interventions, as well as the study of those interventions and their impact (Peterson, [<reflink idref="bib26" id="ref58">26</reflink>]). This method typically involves iterative cycles of design, development, and evaluation, with the aim of generating design principles that can inform future design and development efforts (Paavola et al., [<reflink idref="bib25" id="ref59">25</reflink>]). Overall, design-based research provides a rich and nuanced understanding of the meanings behind preservice teachers' actions, which can inform teacher education and professional development programmes.</p> <p>In implementing design-based research, we established the following conjecture: The use of the graphical calculator in the teaching of functions promotes the development of the didactic knowledge of a preservice mathematics teacher. In realising this conjecture, we outlined the following principles: P1. Prepare lesson plans; P2. Have opportunities for to explore the graphical calculator in the classroom; P3. Reflect on teaching practice.</p> <p>Data collection methods included a teaching practice portfolio (lesson plans (L P) and reflections (R)) and an internship report (I R). The portfolio is a collection of the most relevant activities that the preservice teacher selected to demonstrate their pedagogical practice. The internship report is an academic document of investigative nature about the activities carried out during the preservice teacher's supervised pedagogical practice. With reference to the literature, the aim of these report is to make sense of the teaching and learning activities undertaken during the internship in order to address the research objectives related to the guiding problem of the study.</p> <p>The data analysis method used was content analysis (Bardin, [<reflink idref="bib4" id="ref60">4</reflink>]), taking as reference the dimensions of didactic knowledge. For the data analysis, we utilised the framework based on the work of Ball et al. ([<reflink idref="bib3" id="ref61">3</reflink>]), with the domains and dimensions outlined in Table 1.</p> <p>Table 1. Analysis domains and dimensions.</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Domains MKT</td><td>Dimensions</td></tr></thead><tbody><tr><td>Specialised content knowledge (SCK)</td><td><list list-type="Bullet"><list-item><p>Anticipate alternative representations, provide explanations, and evaluate students' unconventional solution approaches</p></list-item><list-item><p>Identify multiple representations for mathematical problems or concepts</p></list-item><list-item><p>Using multiple representations of mathematical concepts</p></list-item><list-item><p>Identify mathematical concepts and concrete models to justify standard procedures used in the process</p></list-item><list-item><p>Analyse different strategies for solving tasks</p></list-item><list-item><p>Involve students in class activities and discussion of results</p></list-item></list></td></tr><tr><td>Knowledge of content and curriculum (KCC)</td><td><list list-type="Bullet"><list-item><p>Address the curriculum trajectory</p></list-item><list-item><p>Identify teaching materials</p></list-item><list-item><p>Adhere to the methodological guidelines and objectives of the curricular programmes.</p></list-item></list></td></tr><tr><td>Knowledge of content and student (KCS)</td><td><list list-type="Bullet"><list-item><p>Anticipate the contextual factors that support (or hinder) the development of students' understanding.</p></list-item><list-item><p>Anticipate what students will likely do with a task and/or mathematical concepts.</p></list-item><list-item><p>Evaluate the diagnostic potential of tasks or recognise typical student errors.</p></list-item><list-item><p>Understand students' difficulties and interests.</p></list-item><list-item><p>Identify students' potential.</p></list-item></list></td></tr><tr><td>Knowledge content and teaching (KCT)</td><td><list list-type="Bullet"><list-item><p>Anticipate an item's difficulty level and plan mathematical concepts that meet the rigour required by the item.</p></list-item><list-item><p>Describe mathematical tasks or procedures in which students would be involved. Formulate, sequence, and pose questions to students.</p></list-item><list-item><p>Identify strategies to assess students' mathematical understanding.</p></list-item><list-item><p>Identify supplementary resources associated with mathematical topics. Identify different methods and procedures that can be utilised during teaching.</p></list-item><list-item><p>Select and sequence examples or performance tasks that allow students to understand the topic.</p></list-item></list></td></tr></tbody></table> </ephtml> </p> <p>Regarding these dimensions of didactic knowledge, we only consider some of the evidence, presented in the text by E# (# varies from 1 to 27), that emerges in the preservice teacher's discourse from the data collection instruments we used.</p> <hd id="AN0187409024-5">4. Discussion and results</hd> <p>In seeking to understand the meanings that the preservice teacher assigned at different stages of her pedagogical practice, we analysed her discourse in the lesson plans (portfolio), the lesson itself (internship report), and the post-lesson reflection (portfolio) (Table 2).</p> <p>Table 2. Dimensions of analysis about class 1 (Appendix 1).</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Moments</td><td>Evidences</td><td>Dimensions</td></tr></thead><tbody><tr><td>Planning (L <italic>P</italic>)</td><td>'The proper use of graphic calculator can improve the quality of learning' (E1).</td><td>Knowledge of content and curriculum</td></tr><tr><td /><td>'Graphical representations are an intuitive and appealing representation, allowing students to learn the topics under study through visual analysis' (E2).</td><td>Knowledge of content and teaching</td></tr><tr><td /><td>'Students need to develop graphically and analytically what is asked of them. They may have difficulty in solving graphically with the graphic calculator when studying the function' (E3).</td><td>Knowledge of content and students</td></tr><tr><td /><td>'Students may encounter difficulties in using mathematical language on the calculator' (E4).</td><td /></tr><tr><td>Lesson (I R)</td><td>'To determine the intersection of points on the graph of the function, explore the table menus and/or the graph menu. The calculator allows you to determine the points of intersection by defining a neighbourhood of that point. You can also move the cursor to that point and compare the values obtained with those you have determined analytically' (E5).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'To overcome the difficulties of studying the sign of a rational function, compare what you see from the graphical representation generated by the calculator with what you obtain through analytical processes' (E6).</td><td>Knowledge of content and students</td></tr><tr><td>Reflection (R)</td><td>'I had difficult to answer the student who questioned me about a solution using the graphic calculator. I should have prepared better before the lesson. For future lessons, I will try to anticipate the students' questions' (E7).</td><td>Knowledge of content and students</td></tr><tr><td /><td>'I asked the students to first solve the question graphically and then analytically and then compare the two resolutions' (E8).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'Since I did not consider that the students might have difficulty using the calculator, I decided to explain it another way' (E9).</td><td>Knowledge of content and teaching</td></tr></tbody></table> </ephtml> </p> <p>During the planning phase (principle 1), the teacher reveals knowledge of the content and the curriculum by emphasising the methodological recommendations of the curricular programme on the use of the graphing calculator in learning activities: 'the proper use of graphic calculator can improve the quality of learning' (E1). This perspective aligns with Ponte's ([<reflink idref="bib27" id="ref62">27</reflink>]) idea that knowledge of the curriculum and its connections is essential for effective teaching. Moreover, the teacher reveals knowledge of content and teaching by recognising the intuitive and appealing nature of graphical representations obtained on the graphic calculator: 'graphical representations are an intuitive and appealing representation, allowing students to learn the topics under study through visual analysis' (E2). The teacher's approach reflects the concept of 'mathematical knowledge for teaching' developed by Ball et al. ([<reflink idref="bib3" id="ref63">3</reflink>]), which integrates the specialised content knowledge necessary to explain and teach mathematics effectively. By identifying the potential difficulties students may encounter, the teacher also demonstrates an understanding of content and student Knowledge: 'students need to develop graphically and analytically what is asked of them. They may have difficulty in solving graphically with the graphic calculator when studying the function' (E3). This focus on student difficulties is consistent with the emphasis placed by Ponte ([<reflink idref="bib27" id="ref64">27</reflink>]) on understanding students' learning processes and their interactions in the classroom.</p> <p>In the classroom (principle 2), the teacher's actions reflect specialised content knowledge, by explaining how to determine the points of intersection of function graphic, the teacher promoted schemes of instrumented action using the graphing calculator:</p> <p>To determine the intersection of points on the graph of the function, explore the table menus and/or the graph menu. The calculator allows you to determine the points of intersection by defining a neighbourhood of that point. You can also move the cursor to that point and compare the values obtained with those you have determined analytically. (E5)</p> <p>This effort to connect mathematical knowledge with technological tools aligns with the dimensions of didactic knowledge proposed by Ponte ([<reflink idref="bib27" id="ref65">27</reflink>]), which include knowledge of the interrelationships between mathematics and teaching with technologies.</p> <p>The teacher's considerations regarding the students' difficulties in generalising the study of the sign of a rational function through observation on the calculator indicates a knowledge of content and students: 'To overcome the difficulties of studying the sign of a rational function, compare what you see from the graphical representation generated by the calculator with what you obtain through analytical processes' (E6). This type of reflection aligns with the notion that didactic knowledge is built in practice, through interaction and reflection on lived experiences, as advocated by Elbaz ([<reflink idref="bib13" id="ref66">13</reflink>]) and Schön ([<reflink idref="bib31" id="ref67">31</reflink>]).</p> <p>In the reflection phase (principle 3), the teacher recognises the need for better preparation and understanding of the students' difficulties, which demonstrates a commitment to improving knowledge of content and teaching, as well as knowledge of content and students: 'I had difficult to answer the student who questioned me about a solution using the graphic calculator. I should have prepared better before the lesson. For future lessons, I will try to anticipate the students' questions' (E7). In addition, by encouraging students to solve tasks graphically first and then analytically, the teacher applies specialised content knowledge to improve understanding: 'I asked the students to first solve the question graphically and then analytically and then compare the two resolutions' (E8). Finally, the reflection on the need to consider the students' difficulties with using the calculator demonstrates an effort to adapt teaching methods for better results: 'Since I did not consider that the students might have difficulty using the calculator, I decided to explain it another way' (E9). The problematization and reconstruction of moments in their practice promotes the development of their didactic knowledge, as highlighted by Ponte ([<reflink idref="bib27" id="ref68">27</reflink>]), who emphasises the importance of reflection and feedback in the construction of teachers' professional knowledge.</p> <p>From this initial reflection of the future teacher, there is a concern in identifying the students' difficulties, and to overcome them, she recognises the need to adopt new strategies involving the use of graphic calculator, encouraging students to explore this technology. By adjusting her approach based on the observed difficulties, the teacher is committed to adapting her teaching strategies to better support the student's learning in the class 2 (Table 3).</p> <p>Table 3. Dimensions of analysis about class 2 (Appendix 2).</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Moments</td><td>Evidences</td><td>Dimensions</td></tr></thead><tbody><tr><td>Planning (L <italic>P</italic>)</td><td>'To determine the limit of functions through graphical observation, it is expected that students will take advantage of the capabilities of the graphic calculator' (E10).</td><td>Knowledge of the content and students</td></tr><tr><td /><td>'The use of the graphic calculator can improve the understanding of the limit, according to Heine, which is an abstract topic' (E11).</td><td>Knowledge of content and teaching</td></tr><tr><td /><td>'Using the graphic calculator, the students will make conjectures through the observation of graphs' (E12).</td><td>Knowledge of content and curriculum</td></tr><tr><td>Lesson (I R)</td><td>'To determine the limit of functions through graphical observation, must explore the capabilities of the graphing calculator. As you move the cursor closer to the point under study, see what happens to the succession of images that come close to the ordinate of that point' (E13).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'The graphical sketches of the function remain imprecise and show a lack of attention in transcribing what you observe on the calculator to the notebook' (E14).</td><td /></tr><tr><td>Reflection (R)</td><td>'I alerted the students that when creating a graphical sketch, it is necessary to identify and label the coordinate axes and the coordinates of some important points (...). I should also have mentioned that they should always indicate the viewing window they used on the calculator' (E15).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'Despite utilising the potential of the graphic calculator in their learning activities, the students do not adequately justify its use' (E16).</td><td>Knowledge of content and students</td></tr><tr><td /><td>'I presented a student's graphical sketch that did not have a very clear scale, causing the determination of the limit to be misleading. I used this opportunity to alert the students that the graphical representation should be as complete as possible to obtain correct answers' (E17).</td><td /></tr></tbody></table> </ephtml> </p> <p>In the planning phase (principle 1), the teacher reveals an understanding of how the graphic calculator can assist her teaching in understanding abstract mathematical concepts such as limits: 'to determine the limit of functions through graphical observation, (...) students will take advantage of the capabilities of the graphic calculator' (E10). This observation highlights the teacher's knowledge of content and students, in line with the idea that didactic knowledge involves the ability to translate abstract mathematical content into accessible pedagogical practices (Ball et al., [<reflink idref="bib3" id="ref69">3</reflink>]; Ponte, [<reflink idref="bib27" id="ref70">27</reflink>]). Furthermore, by recognising that 'the use of the graphic calculator can improve the understanding of the limit, according to Heine, which is an abstract topic' (E11), the teacher demonstrates an understanding of content and curriculum knowledge. The anticipation that 'the students will make conjectures through the observation of graphs' (E12) reflects the teacher's knowledge of content and students, showing an awareness of how students interact with and understand graphical data.</p> <p>During the lesson (principle 2), the teacher's actions and observations illustrate her specialised content knowledge. The teacher explains that to determine the limit of functions through graphical observation, [the students] 'must explore the capabilities of the graphic calculator. As you move the cursor closer to the point under study, see what happens to the succession of images that come close to the ordinate of that point' (E13). The teacher also observes that 'the graphical sketches of the function remain imprecise and show a lack of attention in transcribing what you observe on the calculator to the notebook' (E14), which indicates an understanding of the students' misconceptions. This point highlights the importance of anticipating learning difficulties in the action and adapting teaching practices to promote a more accurate understanding of mathematical concepts (Ponte, [<reflink idref="bib27" id="ref71">27</reflink>]; Viseu & Menezes, [<reflink idref="bib36" id="ref72">36</reflink>]). Furthermore, guiding students in determining the domain and limit of the function, which reflects specialised content knowledge.</p> <p>In the reflection phase (principle 3), the teacher critically evaluates her teaching methods and the students' results. The teacher's awareness that students need to 'identify and label the coordinate axes and the coordinates of some important points' (E15) She demonstrates a specialised content knowledge. This reflection is in line with the idea that the development of didactic knowledge requires constant revision and adaptation of teaching strategies, taking into account student responses and the demands of the curriculum (Ponte, [<reflink idref="bib27" id="ref73">27</reflink>]; Shulman, [<reflink idref="bib33" id="ref74">33</reflink>]). The teacher reflects on the need for students to 'always indicate the viewing window they used on the calculator' (E15), demonstrating an attention to detail and clarity in the graphical representation. Recognising the potential of the graphic calculator, but noting that 'students do not adequately justify its use' (E16) reveals an ongoing commitment to improving teaching strategies and student understanding.</p> <p>In the preservice teacher's second reflection, she was able to anticipate the students' difficulties, which reinforces the importance of preparing lessons in such a way as to foresee possible obstacles to learning (Ball et al., [<reflink idref="bib3" id="ref75">3</reflink>]; Viseu & Menezes, [<reflink idref="bib36" id="ref76">36</reflink>]). From the insights in the second lesson, the preservice teacher intends to incentive the students in the next lesson to explore the graphic calculator more effectively (Table 4).</p> <p>Table 4. Dimensions of analysis about class 3 (Appendix 3).</p> <p> <ephtml> <table><thead valign="bottom"><tr><td>Moments</td><td>Evidences</td><td>Dimensions</td></tr></thead><tbody><tr><td>Planning (L <italic>P</italic>)</td><td>'The use of the graphic calculator is an excellent tool for determining the asymptotes of a function's graph, as it allows for visualisation and, consequently, a better understanding of the topic' (E18)</td><td>Knowledge of content and teaching</td></tr><tr><td /><td>'I paid attention to the selection of tasks to implement exploratory teaching using the graphic calculator' (E19).</td><td>Knowledge of content and students</td></tr><tr><td /><td>'Students may have difficulties to enter in the calculator an expression such as <p><graphic href="tmes_a_2492779_ilm0001.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">a</mi><mo xmlns="">+</mo><mi xmlns="">b</mi><mrow xmlns=""><mo>/</mo></mrow><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo xmlns="">−</mo><mi xmlns="">c</mi><mo stretchy="false" xmlns="">)</mo></math></p>, since the algorithm for dividing polynomials was mentioned in 10th grade and they may not know how to apply it' (E20).</td><td /></tr><tr><td /><td>'To sketch the graphs of functions, the students rely on usage schemes and instrumented usage schemes of the graphic calculator by editing the expressions that represent the functions and choosing the appropriate viewing window for the graph of each function' (E21)</td><td /></tr><tr><td>Lesson (I R)</td><td>'As some students did not know how to determine the asymptotes, I provided a screenshot from the graphic calculator and indicated the viewing window. A student identified the asymptotes of the function's graph through graphical observation' (E22).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'Gradually, you must take advantage of the graphic calculator's potential to study the asymptotes on the graph of a rational function' (E23).</td><td>Knowledge of content and teaching</td></tr><tr><td>Reflection (R)</td><td>'The students demonstrate appropriating the potential of the graphic calculator in generalising the study of asymptotes to the graphs of rational functions' (E24).</td><td>Knowledge of content and students</td></tr><tr><td /><td>'I spent more time clarifying doubts and tried to explain with various examples so that the students wouldn't make unnecessary mistakes' (E25).</td><td>Specialised content knowledge</td></tr><tr><td /><td>'Students thought that vertical asymptotes were always [lines with] points that did not belong to the domain, but this is not always the case, and they cannot simply justify that <p><graphic href="tmes_a_2492779_ilm0002.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">x</mi><mo xmlns="">=</mo><mi xmlns="">a</mi></math></p> is a vertical asymptote because <p><graphic href="tmes_a_2492779_ilm0003.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">a</mi></math></p> is not in the domain. That's why I felt the need to clarify this issue with the students during the final moments of the lesson' (E26).</td><td>Knowledge of content and teaching</td></tr><tr><td /><td>'Some students attribute processes of instrumented action to the graphic calculator, giving meaning to what they extract from using this tool in their activities' (E27).</td><td /></tr></tbody></table> </ephtml> </p> <p>In this third moment, the students were encouraged to explore the graphic calculator, promoting a critical evaluation of the results obtained when using the calculator. During the planning phase (principle 1), the teacher shows a knowledge of content and teaching by emphasising the value of using the graphic calculator to determine asymptotes on the graphs of functions: 'the use of the graphic calculator is an excellent tool for determining the asymptotes of a function's graph, as it allows for visualisation and, consequently, a better understanding of the topic' (E18). This approach is in line with the idea that a teacher's didactic knowledge includes the ability to explore technologies in order to promote students' understanding (Ponte, [<reflink idref="bib27" id="ref77">27</reflink>]). The teacher's knowledge of content and teaching is evident in the careful selection of tasks designed to encourage exploratory learning: 'I paid attention to the choice of tasks to implement exploratory teaching using the graphing calculator' (E19). This emphasis on task selection reflects the need for detailed planning, as highlighted by Ball et al. ([<reflink idref="bib3" id="ref78">3</reflink>]), and is essential for maximising the effectiveness of technological tools in teaching. By recognising the potential difficulties students may face, such as with division algorithms or the transition from algebraic to graphical representations, the teacher highlights her knowledge of content and students: 'Students may have difficulties to enter in the calculator an expression such as</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></math> </ephtml> , since the algorithm for dividing polynomials was mentioned in 10th grade and they may not know how to apply it' (E20). These difficulties seem to be due not only to a lack of understanding of the algorithm for dividing polynomials, but above all to the missing of brackets between</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>−</mo><mi>c</mi></math> </ephtml> when digitising expressions of the type</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>+</mo><mi>b</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></math> </ephtml> , which is a typing issue.</p> <p>This anticipation of difficulties is in line with the importance of anticipating challenges and adapting instruction to meet students' needs, as advocated by Ponte and Chapman ([<reflink idref="bib29" id="ref79">29</reflink>]).</p> <p>In the classroom, the teacher demonstrates specialised content knowledge by using a screenshot of the graphic calculator to help students determine the asymptotes of a function graph: 'as some students still didn't know how to determine the asymptotes, I provided a screenshot from the graphic calculator and indicated the viewing window' (E22). The gradual appropriation of the graphic calculator's potential by the students reflects the teacher's ability to facilitate the integration of technology into learning: 'gradually, you must take advantage of the graphic calculator's potential to study the asymptotes on the graph of a rational function' (E23). This effective integration of technological tools is consistent with the development of the teacher's didactic knowledge, as mentioned by Boero et al. ([<reflink idref="bib5" id="ref80">5</reflink>]).</p> <p>In the reflection phase (principle 3), the teacher demonstrates knowledge of content and students by identifying how the students generalise the use of the graphic calculator in the study of asymptotes: 'The students demonstrate appropriating the potential of the graphic calculator in generalising the study of asymptotes to the graphs of rational functions' (E24).</p> <p>The teacher's commitment to improving teaching strategies is evident in the fact that she spends time clarifying doubts and uses various examples to avoid mistakes, showing specialised content knowledge: 'I spent more time clarifying doubts and tried to explain with various examples so that the students wouldn't make unnecessary mistakes' (E25). Finally, the teacher's knowledge of content and students is evidenced by addressing misconceptions about vertical asymptotes, ensuring that students understand that they cannot simply justify them on the basis of exclusion from the domain:</p> <p>Students thought that vertical asymptotes were always [lines with] points that did not belong to the domain, but this is not always the case, and they cannot simply justify that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>a</mi></math> </ephtml> is a vertical asymptote because</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> </ephtml> is not in the domain. That's why I felt the need to clarify this issue with the students during the final moments of the lesson. (E26)</p> <p>This attention to detail and clarity in the graphical representation is in line with the role of didactic knowledge in teaching practice, as highlighted by Goldin ([<reflink idref="bib16" id="ref81">16</reflink>]).</p> <p>In the justification provided for the decisions made in the planning of the third lesson, in addition to anticipating student difficulties, the preservice teacher highlights the importance of selecting and designing tasks that enhance the use of the graphic calculator (principle 1 and principle 2) by students, helping them to overcome their difficulties. During lesson 3, it was observed that 'some students attribute processes of instrumented action to the graphic calculator, giving meaning to what they extract from using this tool in their activities' (E27). This observation emphasises how the teacher's specialised content knowledge in integrating the graphic calculator into learning activities enables students to develop a more meaningful understanding of the concepts studied. This evidence can be reflected in the effort the preservice teacher makes to engage students in classroom activities.</p> <hd id="AN0187409024-6">5. Final considerations</hd> <p>This study seeks to understand the contribution of the graphic calculator in the development of didactic knowledge of a preservice teacher in teaching functions to 11th grade students. During the lesson planning phase, the preservice teacher follows the methodological guidelines of the mathematics curriculum, specifically regarding the emphasis on student activity and the connection between different representations of the concepts under study, through the exploration of the graphic calculator, with the aim of promoting students' understanding of the concepts. Such indicators promote the development of curriculum knowledge (Ball et al., [<reflink idref="bib3" id="ref82">3</reflink>]; Ponte & Chapman, [<reflink idref="bib29" id="ref83">29</reflink>]). Gradually, the preservice teacher anticipates possible student difficulties in exploring the graphic calculator when solving tasks, and in articulating the formal mathematical language with the information they retrieve from the graphic calculator. Identifying such difficulties challenged the preservice teacher to plan effective strategies to overcome them through the use of the graphic calculator. These results highlight the relevance, in the lesson planning phase, of not only using the graphic calculator but also anticipating students' difficulties and predicting possible strategies to solve the proposed tasks using the graphic calculator. The creation of teaching strategies that challenge students to give mathematical meaning to the information they retrieve from the graphic calculator underscores the importance of reading specialised texts, such as those on instrumental genesis.</p> <p>In the classroom context, when aiming for students to give mathematical meaning to the information they extract from the graphic calculator, the preservice teacher promotes the concept of instrumental genesis, which is part of specialised teacher knowledge (Kuntze, [<reflink idref="bib19" id="ref84">19</reflink>]). The need for specific and reflective teacher education to integrate technological tools in teaching is thus reinforced in initial teacher education settings.</p> <p>After their pedagogical action, the preservice teacher reflects on situations from their practice by recognising that they do not adequately respond to students' questions regarding task-solving with the graphic calculator, becoming aware of the importance of anticipating, during lesson planning, students' questions, doubts, and solution strategies for future lessons. This retrospective reflection on their practice allows the preservice teacher to reconstruct certain moments of their action, thereby building their practical knowledge (Appova & Taylor, [<reflink idref="bib1" id="ref85">1</reflink>]) on how to teach content using the graphic calculator.</p> <p>This study contributes to the development of knowledge on how to explore the graphic calculator in initial teacher education courses for teaching functions in Mathematics. A central element of this knowledge are the design principles reached through this research. These principles show how future teachers can develop knowledge about the specificities of their actions in exploring the graphic calculator, specialised content knowledge, student work, and the curriculum: P1.1. Promoting lesson planning that explore the use of the graphic calculator; P1.2. Anticipate student difficulties and predict possible strategies for task resolution; P1.3. Promote the reading of texts on instrumental genesis to encourage the exploration of the graphic calculator in a way that gives meaning to the mathematical concepts under study; P2. Create opportunities to explore the graphic calculator in the classroom; P3. Foster reflection on practice.</p> <p>Such principles support the initial conjecture, highlighting other indicators of didactic knowledge, such as: instrumental genesis; the need to anticipate students' difficulties; and the need to anticipate possible strategies for task resolution. The symbiosis between these dimensions of the teacher's knowledge enhances the development of specialised content knowledge to integrate the graphic calculator into teaching practice. This knowledge includes not only mastery of the mathematical content but also the ability to adapt it to students' needs, using technological tools effectively. Observing students assigning meaning to actions facilitated by the graphic calculator underscores the importance of integrating technology in a way that students use it as a means to construct knowledge (Goos, [<reflink idref="bib17" id="ref86">17</reflink>]; Trouche, [<reflink idref="bib35" id="ref87">35</reflink>]).</p> <p>Regarding the anticipation of students' difficulties during lesson preparation, the preservice teacher becomes aware of the relevance of this activity in building their knowledge of both content and students (Ball et al., [<reflink idref="bib3" id="ref88">3</reflink>]). The preservice teacher thus emphasises the importance of predicting potential learning obstacles in order to plan effective teaching strategies (Martins et al., [<reflink idref="bib21" id="ref89">21</reflink>]).</p> <p>Regarding the anticipation of possible task-solving strategies, the results show that the preservice teacher demonstrates concern in predicting possible solutions that facilitate the flow of the lesson and improve student support. This concern is supported by Ponte and Chapman ([<reflink idref="bib29" id="ref90">29</reflink>]), who argue that didactic knowledge should be action-oriented, taking into account possible student responses and the necessary strategies to effectively support them throughout the learning process.</p> <p>The results of this study, which recorded significant progress in the dimensions that structure the preservice teacher's didactic knowledge as a result of the confrontation between lesson planning and its implementation in the classroom context, during moments of reflection on pedagogical practice, lead us to conclude the importance of providing preservice teachers with formative experiences that include the opportunity to teach from an early stage. The positivist model that inspires sequential initial teacher education models, in which professional practice must be preceded by theory (in this case, didactic knowledge in its various aspects) and presented in a formal manner, is questioned by this study, both in the development of teaching practice knowledge and, above all, in the specialised knowledge of the preservice teacher.</p> <hd id="AN0187409024-7">Acknowledgement</hd> <p>This work is funded by CIEd Research Center on Education, Institute of Education, University of Minho, projects UIDB/01661/2020 and UIDP/01661/2020, through national funds of FCT/MCTES-PT; and this work was financially supported by National Funds through FCT-Fundação para a Ciência e a Tecnologia, I. P., under the project UIDB/00194/2020.</p> <hd id="AN0187409024-8">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <hd id="AN0187409024-9">Data availability statement</hd> <p>The data that support the findings of this study are openly available at https://repositorium.sdum.uminho.pt/bitstream/1822/77993/1/Relat%c3%b3rio%2bde%2bEst%c3%a1gio_Ana%20Maria%20de%20Sousa%20Freitas.pdf</p> <hd id="AN0187409024-10">Appendices</hd> <p></p> <hd id="AN0187409024-11">Appendix 1</hd> <p> <bold>Topic:</bold> Rational functions.</p> <p> <bold>Objectives:</bold> Determine the sign of a rational function. Solve fractional inequalities.</p> <p> <bold>Teaching method:</bold> Exploratory teaching.</p> <hd1 id="AN0187409024-12">Fractional equations and the sign of a rational function</hd1> <p></p> <ulist> <item> Consider the functions</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> defined, respectively, by:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mi>x</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></math> </ephtml> .</p> <p></p> <ulist> <item> Sketch, in a single orthogonal reference frame, the graphs of the functions</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> . Mark points <emph>A</emph> and <emph>B</emph> on your sketch, knowing that <emph>A</emph> is the point of intersection of the graphs in the 3rd quadrant and point B is the point of intersection of the graphs in the 1st quadrant. Analytically and graphically study the sign of the function <emph>f</emph>.</p> <p></p> <ulist> <item> Solve, analytically and graphically, the inequality</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>></mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> .</p> <p></p> <ulist> <item> Consider the function <emph>f</emph> defined by</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> , where <emph>P</emph> and <emph>Q</emph> are polynomials e</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Q</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>≠</mo><mn>0</mn></math> </ephtml> . Study the sign of the function <emph>f</emph>.</p> <hd1 id="AN0187409024-13">Practice</hd1> <p></p> <ulist> <item> Solve the following inequality analytically and graphically:</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo><=</mo><mn>1</mn><mrow><mo>/</mo></mrow><mn>2</mn><mi>x</mi><mo>−</mo><mn>1</mn></math> </ephtml> </p> <p></p> <ulist> <item> Study, analytically and graphically, the sign of the function defined by:</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>−</mo><mn>7</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>−</mo><mn>6</mn><mrow><mo>/</mo></mrow><mi>x</mi></math> </ephtml> </p> <p></p> <ulist> <item> The Cartesian referential shows a rectangle [<emph>OAPB</emph>], where the vertex <emph>P</emph>, with a positive abscissa, belongs to the graph of the function <emph>f</emph> defined by</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>6</mn><mi>x</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></math> </ephtml> .</p> <p>Graph</p> <p>If <emph>x</emph> is the x-coordinate of point <emph>P</emph>, determine its value analytically and graphically so that:</p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mrow><mi>OAPB</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> is a square;</p> <p></p> <ulist> <item> The area of the rectangle [<emph>OAPB</emph>] is less than 8.</item> </ulist> <hd1 id="AN0187409024-14">Challenge</hd1> <p>Catarina always drives to school, leaving home between half past seven and eight in the morning. Suppose that when Catherine leaves the house <emph>t</emph> minutes after half past seven, the time of the journey, in minutes, is given by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>45</mn><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>7900</mn></mrow><mrow><mrow><msup><mi>t</mi><mn>2</mn></msup></mrow><mo>+</mo><mn>300</mn></mrow></mfrac></math> </ephtml> </p> <p></p> <ulist> <item> Characterise the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> .</p> <p></p> <ulist> <item> Study the sign of the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> </ephtml> .</p> <p></p> <ulist> <item> Justify the following sentence: 'If Catarina leaves home at 7.40am, she'll arrive at 8.11am, but if she leaves home at 7.55am, she'll be late for class'.</item> <p></p> <item> Using the graphing calculator, determine how late Catherine can leave the house so that she won't be late for school.</item> </ulist> <p>Your resolution should state:</p> <p></p> <ulist> <item> An expression that expresses the problem in question.</item> <p></p> <item> The graph sketch in the context of the problem and the windows used in the graphic calculator.</item> <p></p> <item> The answer to the problem, graphically, adequately justified in hours and minutes.</item> </ulist> <hd id="AN0187409024-15">Appendix 2</hd> <p> <bold>Topic:</bold> Heine limit of real functions of real variable.</p> <p> <bold>Objectives:</bold> Operate with limits of real functions of real variable at adherent points of their domains. Define the product of a bounded function by a function with a zero limit at adherent points in its domain.</p> <p> <bold>Teaching method:</bold> Exploratory teaching.</p> <p> <bold>Task: Operations with limits between real functions of real variable at points adherent to their domains</bold> Consider the real functions of real variable</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> defined, respectively, by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>−</mo><mn>4</mn><mi>x</mi><mo>;</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>2</mn></math> </ephtml> </p> <p></p> <ulist> <item> Complete the following table:</item> <p></p> </ulist> <p> <ephtml> <table><thead valign="bottom"><tr><td>Function</td><td>Graphic sketch</td><td>Determine:</td></tr></thead><tbody><tr><td><p><graphic href="tmes_a_2492779_ilm0028.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">f</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0029.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">Df</mi><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0030.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mi xmlns="">f</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0031.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">g</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0032.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">Dg</mi><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0033.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mi xmlns="">g</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0034.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0035.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>D</mi><mrow><mi>f</mi><mo>+</mo><mi>g</mi></mrow></msub></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0036.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>+</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0037.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0038.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>D</mi><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow></msub></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0039.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>−</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0040.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>×</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0041.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>D</mi><mrow><mi>f</mi><mo>×</mo><mi>g</mi></mrow></msub></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0042.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mo stretchy="false" xmlns="">(</mo><mrow xmlns=""><mi>f</mi><mo>×</mo><mi>g</mi></mrow><mo stretchy="false" xmlns="">)</mo><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0043.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo>(</mo><mrow><mfrac><mi>f</mi><mi>g</mi></mfrac></mrow><mo>)</mo></mrow><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0044.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>D</mi><mrow><mfrac><mi>f</mi><mi>g</mi></mfrac></mrow></msub></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0045.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>1</mn><mo>−</mo></msup></mrow></mrow></munder><mo xmlns="">⁡</mo><mrow xmlns=""><mo>(</mo><mrow><mfrac><mi>f</mi><mi>g</mi></mfrac></mrow><mo>)</mo></mrow><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p><p><graphic href="tmes_a_2492779_ilm0046.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>1</mn><mo>+</mo></msup></mrow></mrow></munder><mo xmlns="">⁡</mo><mrow xmlns=""><mo>(</mo><mrow><mfrac><mi>f</mi><mi>g</mi></mfrac></mrow><mo>)</mo></mrow><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0047.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><mo stretchy="false">[</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msup><mo stretchy="false">]</mo><mn>2</mn></msup></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo><mrow xmlns=""><mspace width="thinmathspace" /></mrow></math></p><p><graphic href="tmes_a_2492779_ilm0048.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow xmlns=""><msub><mi>D</mi><mi>g</mi></msub></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td><td><p><graphic href="tmes_a_2492779_ilm0049.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><munder xmlns=""><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mn>1</mn></mrow></munder><mo xmlns="">⁡</mo><mrow xmlns=""><mo stretchy="false">(</mo><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr></tbody></table> </ephtml> </p> <p></p> <ulist> <item> Consider the real function of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>5</mn><mo>+</mo><mn>1</mn><mrow><mo>/</mo></mrow><mi>x</mi></math> </ephtml> </p> <p></p> <ulist> <item> Determines</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mo stretchy="false">[</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn></mrow><mo stretchy="false">]</mo></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mo stretchy="false">[</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mn>4</mn></mrow><mo stretchy="false">]</mo><mo>;</mo><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mi>x</mi></math> </ephtml> e</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msup><mo stretchy="false">)</mo><mn>3</mn></msup></mrow></math> </ephtml> .</p> <p></p> <ulist> <item> Consider</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> two real functions of real variable, <emph>b</emph> and <emph>c</emph> two real numbers such that</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mi>a</mi></mrow></munder><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mi>a</mi></mrow></munder><mo>⁡</mo><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>c</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> </ephtml> point adherent to the domains of the functions</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> or equal to</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mi mathvariant="normal">∞</mi><mo>.</mo></math> </ephtml> What relation do you see between the limits of the operations between functions</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> ?</p> <hd1 id="AN0187409024-16">Practice</hd1> <p></p> <ulist> <item> Consider the real functions of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> defined, respectively, by</p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mroot><mrow><mi>x</mi><mo>−</mo><mn>2</mn></mrow><mn>3</mn></mroot><mo>;</mo><mspace width="thinmathspace" /><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"><mtr><mtd><mn>4</mn></mtd><mtd><mi>se</mi><mspace width="thinmathspace" /><mi>x</mi><mo><</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mi>se</mi><mspace width="thinmathspace" /><mi>x</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow><mo>−</mo><mn>3</mn><mi>x</mi></mtd><mtd><mi>se</mi><mspace width="thinmathspace" /><mi>x</mi><mo>></mo><mn>1</mn></mtd></mtr></mtable></mrow><mo fence="true" stretchy="true" symmetric="true" /></mrow></math> </ephtml> </p> <p>Determine, by observing the graphs of the functions</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> </ephtml> :</p> <hd1 id="AN0187409024-17">(a)</hd1> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>3</mn><mo>−</mo></msup></mrow></mrow></munder><mo>⁡</mo><mo stretchy="false">(</mo><mrow><mi>h</mi><mo>+</mo><mi>s</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>;</mo></math> </ephtml> (<bold>b)</bold></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>3</mn><mo>+</mo></msup></mrow></mrow></munder><mo>⁡</mo><mo stretchy="false">(</mo><mrow><mi>h</mi><mrow><mo>/</mo></mrow><mi>s</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> ; (<bold>c)</bold></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>1</mn><mo>−</mo></msup></mrow></mrow></munder><mo>⁡</mo><mo stretchy="false">(</mo><mrow><mi>s</mi><mrow><mo>/</mo></mrow><mi>h</mi></mrow><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> ; (<bold>d)</bold></p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mrow><msup><mn>1</mn><mo>−</mo></msup></mrow></mrow></munder><mo>⁡</mo><mrow><mo>(</mo><mrow><msqrt><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></msqrt></mrow><mo>)</mo></mrow></math> </ephtml> </p> <p></p> <ulist> <item> Consider the real function of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mo>+</mo><mn>1</mn><mrow><mo>/</mo></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup></mrow></math> </ephtml> .</p> <p></p> <ulist> <item> Sketch the graph of the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>;</mo></math> </ephtml> </p> <p></p> <ulist> <item> Sketch and determine a possible real function of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> </ephtml> , so that:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>(</mtext></mrow><mrow><msub><mi mathvariant="bold-italic">b</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>(</mtext></mrow><mrow><msub><mi mathvariant="bold-italic">b</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>+</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>×</mo><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>+</mo><mi mathvariant="normal">∞</mi></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mtext>(</mtext></mrow><mrow><msub><mi mathvariant="bold-italic">b</mi><mn>3</mn></msub></mrow><mo stretchy="false">)</mo><munder><mrow><mo form="prefix">lim</mo></mrow><mrow><mi>x</mi><mo stretchy="false">→</mo><mo>−</mo><mi mathvariant="normal">∞</mi></mrow></munder><mo>⁡</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></math> </ephtml> </p> <p></p> <ulist> <item> Justify, using a graphic calculator, that</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>x</mi><mo stretchy="false">→</mo><msup><mn>0</mn><mo>+</mo></msup></mrow></munder><mrow><mo>(</mo><mi>xcos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mrow><mo>/</mo></mrow><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><msqrt><mi>x</mi></msqrt><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math> </ephtml> .</p> <hd id="AN0187409024-18">Appendix 3</hd> <p> <bold>Topic:</bold> Asymptotes to the graph of a real function of real variable.</p> <p> <bold>Objective:</bold> Determine the asymptotes of the Cartesian graph of rational functions defined in</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mi mathvariant="double-struck">R</mi></mrow></mrow><mi mathvariant="normal">∖</mi><mo fence="false" stretchy="false">{</mo><mi>c</mi><mo fence="false" stretchy="false">}</mo></math> </ephtml> by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></math> </ephtml> .</p> <p> <bold>Teaching method:</bold> Exploratory teaching.</p> <hd1 id="AN0187409024-19">Task: Asymptotes to the graph of rational functions defined by expressions of the type</hd1> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></math> </ephtml> </p> <p>Consider the real functions of real variable</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>,</mo><mi>g</mi><mo>,</mo><mi>h</mi></math> </ephtml> and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math> </ephtml> defined, respectively, by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>6</mn><mi>x</mi><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo></math> </ephtml> ;</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>4</mn><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>5</mn><mo stretchy="false">)</mo></math> </ephtml> </p> <p></p> <ulist> <item> Complete the following table:</item> <p></p> </ulist> <p> <ephtml> <table><thead valign="bottom"><tr><td>Function defined by an expression of the form <p><graphic href="tmes_a_2492779_ilm0090.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">a</mi><mo xmlns="">+</mo><mi xmlns="">b</mi><mrow xmlns=""><mo>/</mo></mrow><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo xmlns="">−</mo><mi xmlns="">c</mi><mo stretchy="false" xmlns="">)</mo></math></p></td><td>Graphic sketch</td><td>Domain</td><td>Vertical asymptotes</td><td>Horizontal asymptotes</td><td>Monotony</td></tr></thead><tbody><tr><td><p><graphic href="tmes_a_2492779_ilm0091.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">f</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0092.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">g</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0093.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">h</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr><tr><td><p><graphic href="tmes_a_2492779_ilm0094.gif" content-type="Graph" /><math xmlns="http://www.w3.org/1998/Math/MathML"><mi xmlns="">i</mi><mo stretchy="false" xmlns="">(</mo><mi xmlns="">x</mi><mo stretchy="false" xmlns="">)</mo><mo xmlns="">=</mo><mo xmlns="">...</mo></math></p></td></tr></tbody></table> </ephtml> </p> <p></p> <ulist> <item> Consider a real function of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo></math> </ephtml> , with</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi></math> </ephtml> e</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> </ephtml> real numbers, determines the vertical and horizontal asymptotes to the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math> </ephtml> .</p> <hd1 id="AN0187409024-20">Practice</hd1> <p></p> <ulist> <item> Determine the asymptotes to the graph of the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo>−</mo><mn>8</mn><mo stretchy="false">)</mo></math> </ephtml> and sketch the graph and its asymptotes.</p> <p></p> <ulist> <item> About the Cartesian graph of the real function of real variable</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>20</mn><mo>−</mo><mn>10</mn><mi>x</mi><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mn>3</mn><mo stretchy="false">)</mo></math> </ephtml> we know that:</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi></math> </ephtml> is the horizontal asymptote and</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>b</mi></math> </ephtml> is the vertical asymptote to the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> ;</p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> is the point of intersection of the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> with the axis</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ox</mi></math> </ephtml> ;</p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> </ephtml> is the point of intersection of the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> with the axis</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Oy</mi></math> </ephtml> ;</p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math> </ephtml> is the point of intersection of the asymptotes y =</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> </ephtml> e</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>b</mi><mo>;</mo></math> </ephtml> </p> <p></p> <p>• </p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> </ephtml> is the intersection point of the asymptote</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi></math> </ephtml> with the axis</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Oy</mi></math> </ephtml> ;</p> <p>Make a graphical sketch of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> that takes into account the characteristics presented and determines the area of the quadrilateral</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">[</mo><mrow><mi>ABCD</mi></mrow><mo stretchy="false">]</mo></math> </ephtml> .</p> <p>Challenge</p> <p>Consider a real function of real variable</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>kx</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mrow><mo>/</mo></mrow><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></math> </ephtml> , where</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> </ephtml> is a positive real number. Let <emph>A</emph> be the point of intersection of the asymptotes of the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> .</p> <p></p> <ulist> <item> Show that</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mi>AO</mi></mrow><mo accent="false">¯</mo></mover><mo>=</mo><msqrt><mn>2</mn></msqrt><mi>k</mi></math> </ephtml> , where O is the origin of the Cartesian referential.</p> <p>To answer this question, you must:</p> <p></p> <ulist> <item> Determine the asymptotes to the graph of the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> ;</p> <p></p> <ulist> <item> Draw a graphical sketch of the function</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> </ephtml> indicating the coordinates of the point</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> ;</p> <p></p> <ulist> <item> Determine the distance from the point</item> </ulist> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> </ephtml> to the point</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> </ephtml> .</p> <p></p> <p>• Suppose</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2</mn></math> </ephtml> , graphically and analytically determines the asymptotes to the graph of the function</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> </ephtml> defined by</p> <p>Graph</p> <p> <ephtml> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mrow><mi>x</mi><mo>−</mo><mn>3</mn></mrow><mo stretchy="false">)</mo><mo>×</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></math> </ephtml> .</p> <ref id="AN0187409024-21"> <title> Note </title> <blist> <bibl id="bib1" idref="ref11" type="bt">1</bibl> <bibtext> The Portuguese education system comprises 12 years before entering higher education. 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  Group: Ti
  Data: Using a Graphic Calculator to Promote a Preservice Teacher's Didactic Knowledge of Functions
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  Group: Lang
  Data: English
– Name: Author
  Label: Authors
  Group: Au
  Data: <searchLink fieldCode="AR" term="%22Floriano+Viseu%22">Floriano Viseu</searchLink><br /><searchLink fieldCode="AR" term="%22Ana+Paula+Aires%22">Ana Paula Aires</searchLink><br /><searchLink fieldCode="AR" term="%22Sara+Cruz%22">Sara Cruz</searchLink>
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  Data: <searchLink fieldCode="SO" term="%22International+Journal+of+Mathematical+Education+in+Science+and+Technology%22"><i>International Journal of Mathematical Education in Science and Technology</i></searchLink>. 2025 56(8):1457-1475.
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  Label: Availability
  Group: Avail
  Data: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
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  Data: Y
– Name: Pages
  Label: Page Count
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  Data: 19
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  Label: Publication Date
  Group: Date
  Data: 2025
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  Label: Document Type
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  Data: Journal Articles<br />Reports - Research<br />Tests/Questionnaires
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  Label: Education Level
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  Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Secondary+Education%22">Secondary Education</searchLink><br /><searchLink fieldCode="EL" term="%22Grade+11%22">Grade 11</searchLink><br /><searchLink fieldCode="EL" term="%22High+Schools%22">High Schools</searchLink>
– Name: Subject
  Label: Descriptors
  Group: Su
  Data: <searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teachers%22">Preservice Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Graphing+Calculators%22">Graphing Calculators</searchLink><br /><searchLink fieldCode="DE" term="%22Secondary+School+Teachers%22">Secondary School Teachers</searchLink><br /><searchLink fieldCode="DE" term="%22Preservice+Teacher+Education%22">Preservice Teacher Education</searchLink><br /><searchLink fieldCode="DE" term="%22Grade+11%22">Grade 11</searchLink><br /><searchLink fieldCode="DE" term="%22Knowledge+Level%22">Knowledge Level</searchLink><br /><searchLink fieldCode="DE" term="%22Pedagogical+Content+Knowledge%22">Pedagogical Content Knowledge</searchLink><br /><searchLink fieldCode="DE" term="%22Foreign+Countries%22">Foreign Countries</searchLink><br /><searchLink fieldCode="DE" term="%22Lesson+Plans%22">Lesson Plans</searchLink><br /><searchLink fieldCode="DE" term="%22Planning%22">Planning</searchLink>
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  Data: <searchLink fieldCode="DE" term="%22Portugal%22">Portugal</searchLink>
– Name: DOI
  Label: DOI
  Group: ID
  Data: 10.1080/0020739X.2025.2492779
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  Label: ISSN
  Group: ISSN
  Data: 0020-739X<br />1464-5211
– Name: Abstract
  Label: Abstract
  Group: Ab
  Data: The initial teacher education courses for Mathematics aim to promote an adequate education in mathematics and mathematics didactics. Equally or even more important than this education is understanding how the preservice teacher should build new knowledge from the activities they undertake. Considering the teaching practicum, this study aims to understand the contribution of the graphic calculator that complies with the guidelines set by the Portuguese Ministry of Education for secondary education in promoting the didactic knowledge of a preservice teacher in the teaching of functions in 11th grade within the Portuguese education system. Adopting a qualitative and interpretive methodology, following a design-based approach, data were collected through her lesson plans, reflections and practicum report. Regarding content knowledge of functions, the preservice teacher identifies different representations and reveals knowledge of facts and procedures. In terms of specialised content knowledge, she explores the graphic calculator to connect representations and identifies different task-solving strategies. About content knowledge and the curriculum, she knows the curricular recommendations about the teaching of functions and the integration of the graphic calculator in mathematics lessons. Regarding content knowledge and students, she identifies factors that enhance or inhibit understanding and evaluates students' comprehension of the topics studied.
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  Data: 2025
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  Data: EJ1481510
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      – Type: doi
        Value: 10.1080/0020739X.2025.2492779
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      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 19
        StartPage: 1457
    Subjects:
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Preservice Teachers
        Type: general
      – SubjectFull: Graphing Calculators
        Type: general
      – SubjectFull: Secondary School Teachers
        Type: general
      – SubjectFull: Preservice Teacher Education
        Type: general
      – SubjectFull: Grade 11
        Type: general
      – SubjectFull: Knowledge Level
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      – SubjectFull: Pedagogical Content Knowledge
        Type: general
      – SubjectFull: Foreign Countries
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      – SubjectFull: Lesson Plans
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      – SubjectFull: Planning
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      – SubjectFull: Portugal
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      – TitleFull: Using a Graphic Calculator to Promote a Preservice Teacher's Didactic Knowledge of Functions
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            NameFull: Ana Paula Aires
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            NameFull: Sara Cruz
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            – Type: volume
              Value: 56
            – Type: issue
              Value: 8
          Titles:
            – TitleFull: International Journal of Mathematical Education in Science and Technology
              Type: main
ResultId 1