<p>In this unit, students will create absolute value functions. Students will:<br>- make a table of values and graphs for absolute value functions.<br>- understand how absolute value functions are transformed.<br>- model real-world situations using absolute value functions and analyze their components, including vertex, domain, range, and rate of change.<br>- develop the ability to imagine real-world scenarios that are best represented by absolute value functions.<br>&nbsp;</p>

<p>- How can you represent a given function? What relationships can be drawn between the various representations?<br>- Which function best represents a specific real-world scenario? How does the function manifest itself in the real world?<br>- How may an equation, table, and graph be used to examine the rate of change and other relevant data for a real-world situation and its representative function?</p>

<p>- Absolute Value Function: A function in the form of , where <i>a</i> ≠ 0.<br>- Dependent Variable: The variable representing range values of a function, commonly the <i>y</i>-term.<br>- Domain: The set of <i>x</i>-values or input values of a function.<br>- Independent Variable: The variable representing domain values of a function, commonly the <i>x</i>-term.<br>- Range: The set of <i>y</i>-values or output values of a function.<br>- Vertex: The point, or ordered pair, that represents the minimum or maximum of a function.<br>- Rate of Change: The difference in the change in <i>y</i>-values per change in <i>x</i>-values (e.g., slope).</p>

<p>- Lesson 1 Exit Ticket (M-A2-7-1_Lesson 1 Exit Ticket)<br>- Lesson 1 Graphic Organizer (M-A2-7-1_Lesson 1 Graphic Organizer)<br>- Absolute Value Worksheet (M-A2-7-1_Absolute Value Worksheet)<br>- Internet access for students</p>

<p>- Examine students' reasoning using the hypotheses and examples they provide to determine their degree of comprehension.<br>- During the lesson, assess students' ability to create tables and graphs, as well as define vertex, domain, range, and rate of change. The intricacies of each activity indicate their engagement with the relationship.<br>- Collaborate with partners to create real-world absolute value functions and representations, and design transformation rules. Collaboration and communication indicate a higher degree of understanding.<br>- Use applet exploration (Grapher applet) to measure students' technology proficiency.<br>- Independent work on creating real-world absolute value functions and details, such as vertex (min/max), domain, range, and rate of change, demonstrates the ability to explore beyond the lesson.<br>- Use the Lesson 1 Exit Ticket (M-A2-7-1_Lesson 1 Exit Ticket) with elements tailored to diverse student capacities.<br>&nbsp;</p>

<p>Active Engagement, Modeling&nbsp;<br>W: The implementation of active engagement, discussion, and partner and independent work will disclose student comprehension of absolute value functions and their transformations.<br><br>H: The lesson provides a conceptual foundation for students to flourish by brainstorming and hypothesizing about absolute value functions, as well as real-world scenarios that can be modeled using them.<br><br>E: The lesson is broken into two parts: Part 1 serves as the emphasis, while Part 2 includes two in-depth tasks. Students are encouraged to generate their own ideas, visualize functions and transformations, imagine absolute value functions in the real world before exploring examples, investigate function transformations rather than rote rule memorization, and confirm further understanding through an applet exploration activity. The concluding requirement encompasses all aspects of the lesson, from creating a real-world scenario to interpreting the function's components. The lesson's structure encourages research prior to learning definitions and rules.<br><br>R: The brief presentation on absolute value functions encourages students to reflect, reread, modify, and reconsider with a partner. Students must be aware of and informed about each instructional component in order to create the short presentation. The class discussion focuses on reflection, rethinking, revising, and revisiting.<br><br>E: The lesson's discussion requirements encourage students to communicate their understanding, reflect on others' comments, and re-evaluate their own understanding.<br><br>T: The lesson includes many learning tools. Students, for example, are exposed to a wide range of visual representations, discursive opportunities, explorations, and discoveries. The option to collaborate with a partner at various points of the session is beneficial for students who benefit from conversation and social engagement and/or require additional assistance throughout the lesson.<br><br>O: The class is structured to introduce topics through an inquiring approach, followed by modeling and individual work for students. The lesson stresses conceptual comprehension of absolute value functions as a whole, instructing students to picture such functions in the real world and develop authentic connections when extracting information from them.</p>

<p><strong>Part 1</strong></p><p>Students will form hypotheses about what they think an absolute value function looks like. Ask students to recall the definition of absolute value (the distance a number is from zero). With a function, each input has exactly one output. As a result, if the number 2 is two units away from 0, so −2 is also two units from 0. Thus, both <i>x</i>-values of -2 and 2 result in an output value of 2. Students may recommend creating a value table to analyze the connection.</p><figure class="image"><img style="aspect-ratio:216/364;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_134.png" width="216" height="364"></figure><p>Students should note that the y-values are all positive; hence, the graph will take the shape of a V, opening up. Students should also notice that the origin is at (0, 0). Ask students to plot the ordered pairs using the table above.</p><p>Students should help generate the graph shown below.</p><figure class="image"><img style="aspect-ratio:381/331;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_135.png" width="381" height="331"></figure><p>Explain to students that the points should be connected to display the function.</p><p>Doing so will result in the graph shown below.</p><figure class="image"><img style="aspect-ratio:373/279;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_136.png" width="373" height="279"></figure><p>Take one of the examples and represent it using a table and graph.</p><p>Example problem:</p><p>Joseph is traveling home for the weekend. He wants to calculate the number of miles he is away from home, both prior to his arrival and following his departure, for each hour that passes.</p><figure class="image"><img style="aspect-ratio:598/301;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_137.png" width="598" height="301"></figure><figure class="image"><img style="aspect-ratio:385/311;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_138.png" width="385" height="311"></figure><p><br>It can be noted that the distance decreases, reaches zero, and then increases. The graph decreases steadily from negative 2 hours (or 2 hours before arrival) until arrival at the destination (represented by the origin (0, 0)). There is a constant rate of change. The graph increases steadily from departure from the destination (which is again represented by the origin (0, 0). Again, there is a constant <i>rate of change</i> at 60 for each hour that passes.</p><p>The absolute value function's vertex coincides with the origin (0, 0). In this real-world situation, the vertex, or minimum, relates to the destination (time and mileage are both zero). Joseph is no longer traveling, so there is no time or mileage recorded. The vertex is at (0, 0), indicating that the function is not moved right or left, up or down.</p><p>It is clear that this type of real-world situation should be represented by an <i>absolute value function</i>, which is a nonlinear function.</p><p>Ask students to consider a real-world scenario that may involve an absolute value function.</p><p>Examples could include distance from school and time taken, both walking to and from the school; mileage driven and hours taken, and so on (distance and time problems)</p><p>For example, a student walks one mile to school each morning and one mile home each afternoon, taking 15 minutes each way. If the time before the student arrives at school is -15, what numbers represent the student's arrival at school and home? (<i>0, 15</i>).</p><p>Place each student with a partner and work together to develop a real-world absolute value function, table of values, and graph. Ask students to be as detailed as possible about the graph and table. How do they relate to the problem? How would you describe them in everyday language?</p><p><strong>Part 2</strong></p><p><strong>Absolute Value Parent Function and Variations Thereof</strong></p><p>Introduce students to the absolute value <i>parent function</i> and the converted absolute value functions. A <i>parent function</i> is a function that has not been transformed.</p><p>See the examples below:</p><p>y = |x|</p><p>y = 2|x|</p><p>y = –|x|</p><p>y = –2|x|</p><p>y = |x| + 2</p><p>y = |x + 2|</p><p>y = |x| – 2</p><p>y = |x – 2|</p><p><br>Have students investigate transformations using tables and graphs. Before creating each representation, have students consider what will happen to the output and graph with each modified equation.</p><p>For each of the above examples, students may predict that the graph is narrower or wider, reflected across one of the axes, and shifted up or down 2. Students may have difficulties grasping the transformation within the absolute value bars, y = |x + 2| and y = |x - 2|. Students may interpret these as upward or downward transformation, or switch the direction of the shift left or right. Students should generate a table of values and a graph for each guess to confirm or reject these guesses.</p><p>Next, ask students to compare the graphs and tables, as well as compare the graphs with the equations. What happened? Why did such a transformation occur? Why did y = |x + 2| result in a shift of the parent function two units to the left? Why did y = |x - 2| result in a shift of the parent function two units to the right?</p><p>To answer these questions, students should compare the tables, graphs, and equations provided.</p><figure class="image"><img style="aspect-ratio:597/717;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_139.png" width="597" height="717"></figure><p>By looking at the output or <i>y</i>-values, students can see that the <i>y</i>-values caused the graph to shift to the left two units for y = |x + 2| and to the right two units for y = |x - 2|. Using y = |x + 2|, the y-values decreased by 2 till the <i>x</i>-value of −1. The y-values increased by 2 after that. With y = |x - 2|, the <i>y</i>-values increased by 2 until the <i>x</i>-value of 1. After that, the <i>y</i>-values decreased by 2. The main point to notice is that the vertex transformed from (0, 0) with y = |x| to (−2, 0) with y = |x + 2| and (2, 0) with y = |x - 2|. As a result, the reasoning behind shifting two units to the left with y = |x + 2| and two units to the right with y = |x - 2|.</p><p>Distribute copies of the Lesson 1 Graphic Organizer (M-A2-7-1_Lesson 1 Graphic Organizer).</p><p>The graphs for each function show the left and right shifts. Note that the lines for abs(<i>x</i> + 2), abs(<i>x</i>), and abs(<i>x</i> - 2) are red, black, and green, respectively.</p><figure class="image"><img style="aspect-ratio:559/395;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_140.png" width="559" height="395"></figure><p><strong>"Let's try to write some rules about transformations of absolute value functions."</strong></p><p>Encourage students to collaborate with a partner to write transformation rules. Discuss the rules with the students and build the chart shown below.</p><p><strong>"When comparing graphs of absolute value functions with the parent absolute value function y = |x| note the following" :</strong></p><figure class="image"><img style="aspect-ratio:598/353;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_141.png" width="598" height="353"></figure><p>Examining a sample of the previous graphs, namely, y = |x|, y = |x + 2|, and y = |x| - 2, let's determine the domain and range of each. Let's also determine the rate of change.</p><p>The <i>domain</i> of a function is the set of <i>x</i>-values, also known as input values. The <i>range</i> of a function is the set of <i>y</i>-values, also known as output values.</p><p>y = |x|</p><figure class="image"><img style="aspect-ratio:216/181;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_142.png" width="216" height="181"></figure><p>The domain of the function y = |x| is all real numbers, or (-∞, ∞).</p><p>The function y = |x| has a range of all non-negative real numbers, which is [0, ∞).</p><p>The graph indicates that the <i>x</i>-values spread infinitely in both directions, including all reals. The graph also demonstrates that the <i>y</i>-values extend infinitely above and include the value of 0.</p><p>y = |x + 2|</p><figure class="image"><img style="aspect-ratio:230/175;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_143.png" width="230" height="175"></figure><p>The domain of the function y = |x + 2| is all real numbers, or (-∞, ∞).</p><p>The range of the function y = |x + 2| is all non-negative real numbers ([0, ∞]).</p><p>The graph indicates that the <i>x</i>-values spread infinitely in both directions, including all reals. The graph also demonstrates that the <i>y</i>-values extend infinitely above and include the value of 0. It should be noted that the domain and range of this modified absolute value function are the same as the parent absolute value function.</p><p>y = |x| – 2</p><figure class="image"><img style="aspect-ratio:246/189;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_144.png" width="246" height="189"></figure><p>The domain of the function y = |x| - 2 includes all real numbers.</p><p>The function y = |x| - 2 covers all real numbers greater than or equal to −2, or [−2, ∞). The graph indicates that the <i>x</i>-values spread infinitely in both directions, including all reals. The graph illustrates an unlimited range of <i>y</i>-values above and including -2.</p><p>Use the Absolute Value Worksheet (M-A2-7-1_Absolute Value Worksheet) to help students practice using absolute value functions.</p><p>"How can the rate of change be determined?"</p><p>The absolute value parent function has the rate of change is -1 up until the origin. The rate of change is then one. Note that rate of change is synonymous with slope.</p><p>For y = |x + 2|, the rate of change is -1 up until the vertex. The rate of change is then 1.</p><p>For y = |x| - 2, the rate of change is -1 up until the vertex. The rate of change is then 1.</p><p>Ask students to discover the domain, range, rate of change, and vertex of other transformed functions.</p><p><strong>What Does a Reflection Look Like?</strong></p><p><strong>“Can we plot an absolute value function on the </strong><i><strong>x</strong></i><strong>-axis? Certainly. This function is represented as y = -|x|. What happens to the </strong><i><strong>y</strong></i><strong>-values under such a transformation? Yes, the </strong><i><strong>y</strong></i><strong>-values are negated. As a result, the graph is reflected across the </strong><i><strong>x</strong></i><strong>-axis. Let's take a look.”</strong></p><figure class="image"><img style="aspect-ratio:337/272;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_145.png" width="337" height="272"></figure><p><br>Note that the graph lines for -abs(<i>x</i>) and abs(<i>x</i>) are purple and orange, respectively.</p><p>Ask students to create a table of values and graph the absolute value functions:</p><p>y = 4|x|</p><p>y = –4|x|</p><p>y = |x – 4|</p><p>y = –|x – 4|</p><p><strong>Activity 2: Exploring with a Virtual Grapher</strong></p><p>Students can experiment with absolute value functions and function transformations using NLVM's virtual Grapher applet, which is available at <a href="http://enlvm.usu.edu/ma/nav/activity.jsp?sid=nlvm&amp;cid=4_2&amp;lid=109">http://enlvm.usu.edu/ma/nav/activity.jsp?sid=nlvm&amp;cid=4_2&amp;lid=109</a>&nbsp;</p><figure class="image"><img style="aspect-ratio:596/271;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_146.png" width="596" height="271"></figure><p>Students should utilize the applet to investigate the parent function y = |x| and transformations, such as reflection across the <i>x</i>-axis, y = -|x|. Students should learn different transformations, such as translations to the right, left, up, and down. Students should record notes and observations about the relationships between the function equation and the resulting graph as they explore and discover with the applet. Encourage students to investigate min/max (vertex), domain, range, and rate of change. Students should also experiment with the applet to see the changes in the width of the function when varying the value of the coefficient of <i>a</i>.</p><p>Students will be asked to imagine an absolute value function that represents a new real-world scenario. They should generate a table of values and a graph to demonstrate the function. Students should describe the rate of change and meaning of the graph's vertex in the context of the problem. They should also consider the meaning of domain and range in the context of the problem.<br><br>For review, encourage students to construct a brief presentation that relates the parent function to other absolute value functions. Encourage students to concentrate on making connections and highlighting the conceptual foundations of dealing with these types of functions. The primary goal of the lesson should be to teach students when modeling with absolute value functions is appropriate, how to extract information from the equation, table, and graph, and how to verbalize the findings in a way that is relevant to the problem's context. Allow time for discussion at the end of the presentations.</p><p>Provide students with the Lesson 1 Exit Ticket (M-A2-7-1_Lesson 1 Exit Ticket) and allocate objects based on their difficulty level.</p><p><strong>Extension:</strong></p><ul><li>Investigate x- and y-intercepts of absolute value functions. In addition, present more challenging functions that use a combination of transformations. Examples are:</li></ul><p>y = -3|x - 6| + 4,&nbsp;<br>y = 2|x + 10|,&nbsp;<br>y = -\(\vert{x}-\frac{1}{2}\vert\) +2, etc.</p><ul><li>Provide instances of absolute value functions that use more difficult numbers, including fractions and decimals.</li></ul>