<p>In this unit, students will create quadratic functions. Students will:<br>- create a table of values and graphs of quadratic functions.<br>- understand how to transform quadratic functions.<br>- use quadratic functions to model real-world situations and interpret components such as vertex, domain, range, and rate of change.<br>- develop the ability to visualize real-world scenarios using quadratic functions.</p>

<p>- How can students 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 can equations, tables, and graphs be utilized to examine the rate of change and other relevant information for a real-world problem or function?</p>

<p>- 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>- Extraneous Solution: A number obtained in the process of solving an equation which is not a root of the equation given to be solved.<br>- Independent Variable: The variable representing domain values of a function, commonly the <i>x</i>-term.<br>- Orientation: The direction of the opening of the graph (e.g., opening up-positive or opening down-negative).<br>- Origin: The point (0, 0) on a graph.<br>- Quadratic Function: A function in the form of <i>y</i> = a<i>x</i>² + c, or <i>y</i> = a<i>x</i>² + b<i>x</i> + c, where <i>a</i> ≠ 0.<br>- Parabola: The shape of a quadratic function.<br>- Range: The set of <i>y</i>-values or output values 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).<br>- Vertex: The point, or ordered pair, that represents the minimum or maximum of a function.</p>

<p>- Lesson 2 Exit Ticket (M-A2-7-2_Lesson 2 Exit Ticket and M-A2-7-2_Lesson 2 Exit Ticket KEY)<br>- Quadratic Modeling Worksheet (M-A2-7-2_Quadratic_Modeling_Worksheet and M-A2-7-2_Quadratic_Modeling_Worksheet_Key)<br>- Lesson 2 Graphic Organizer (M-A2-7-2_Lesson 2 Graphic Organizer)<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 class, assess students' grasp of table and graph formation, including defining vertex, domain, range, orientation, and rate of change.<br>- Observe students collaborate with their partners to identify the domain, range, vertex, and orientation of converted quadratic functions. Collaboration and communication indicate a higher degree of understanding.<br>- Use applet exploration (Grapher applet) to evaluate student engagement with technological applications.<br>- Evaluate students' comprehension of the relationship between the domain, range, and graph of each relation through individual work on matching activities, rule writing, and real-world parabolic situations.<br>- Use the Lesson 2 Exit Ticket (M-A2-7-2_Lesson 2 Exit Ticket and M-A2-7-2_Lesson 2 Exit Ticket KEY) to assess students' comprehension of the function's graphical representation.</p>

<p>Active Participation, Modeling<br>W: Active participation, modeling, discussion, and collaborative and autonomous work will improve students' comprehension of quadratic functions and transformations.<br>H: Predicting quadratic appearances and associated representations, as well as brainstorming real-world scenarios and connections, gives a basic notion with many beginning places and logical paths to build comprehension.<br>E: The lesson is broken into two parts: Part 1 serves as the emphasis or hook, and Part 2 includes detailed tasks. The course concludes with a recap of a top-down method for exploring quadratic functions.<br>R: The lesson encourages students to see and grasp quadratics, parabolas, and their real-world links. The emphasis on developing rules/definitions for each concept explored in the session facilitates introspection, revisiting, modifying, and rethinking.<br>E: Students have adequate time and opportunity to discuss ideas with group and class members, allowing them to re-evaluate their own understanding.<br>T: All students can benefit from a variety of techniques that include conversation, investigation, discovery, real-world scenarios, quadratic function modeling, and peer collaboration.<br>O: The lesson is designed with inquiry, questioning, modeling, and independent/group work. Students are asked to visualize quadratic functions in the actual world and identify when and where they are applied.</p>

<p><strong>Part 1</strong><br>Ask students the following question: <strong>"What do you know about quadratic functions? What do they look like? Where have you encountered quadratic functions? What exactly does 'parabola' mean? How does it relate to a quadratic function? When do you need to use a parabola? Who uses parabolas on a daily basis to solve problems? How can we represent a quadratic function?"</strong></p><p>Students should respond by stating that quadratic functions can be expressed as equations, tables, and graphs. Quadratics can also be described with words.</p><p><strong>"Before we go into the many representations of quadratic functions, let's make sure we understand the terms </strong><i><strong>quadratic function</strong></i><strong> and </strong><i><strong>parabola</strong></i><strong>. A quadratic function is the actual function described in numerous forms. A parabola is the shape of the graph. The parabola is the shape of the graph of a quadratic function."</strong></p><p><strong>"Does anyone have a basic definition of a quadratic function?"</strong></p><p>Students may respond that a quadratic function is not linear and increases or decreases more rapidly from the vertex of the graph.</p><p><strong>"What would the equation look like?"</strong></p><p>Students should respond that the leading word will include a variable of degree two. Examples are: y = 3x², y = -x², y = 12x² + 2, y = x² - 3, and so on.</p><p><strong>"What would the table look like? In other words, how will the change in </strong><i><strong>y</strong></i><strong>-values affect the change in </strong><i><strong>x</strong></i><strong>-values? How does the change in </strong><i><strong>y</strong></i><strong>-values from a quadratic function compare to the change in </strong><i><strong>y</strong></i><strong>-values from a linear function?"</strong></p><p>Students should respond that the change in <i>y</i>-values is greater than the change in <i>x</i>-values. Furthermore, the change in <i>y</i>-values from a quadratic function decreases/increases faster than that from a linear function. A quadratic function does not have a constant rate of change.</p><p><strong>"How would the graph look? What shape would it be?"</strong></p><p>Students should predict the parabola's U-shape. Encourage your students to sketch a parabola that opens up and down. The parabola has a minimum point if it expands up, and a maximum point when it opens down.</p><p><strong>"Now that we have some hypotheses, let's investigate and identify different quadratic function representations and their corresponding parabolas."</strong></p><p><strong>"First, consider the forms of a quadratic function. A quadratic function can be expressed as "y = ax² + bx + c, y = ax² + c."</strong></p><p>Students who create a table of values to analyze the relationship will gain a better understanding of how the function behaves.</p><figure class="image"><img style="aspect-ratio:231/366;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_147.png" width="231" height="366"></figure><p>Students should notice that all of the <i>y</i>-values are positive, resulting in a U-shaped graph that opens up. In other words, the <i>orientation</i> will be positive. Students should also notice that the origin is at (0, 0). Ask students to plot the ordered pairs using the table above. (Compare the table of values for the absolute value parent function to this table, as well as the graphs of a quadratic and an absolute value function.)</p><p>Students should help generate the graph shown below.</p><figure class="image"><img style="aspect-ratio:322/259;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_148.png" width="322" height="259"></figure><p>Discuss with the students how the points should be connected to display the function. Doing so will result in the graph shown below.</p><figure class="image"><img style="aspect-ratio:348/262;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_149.png" width="348" height="262"></figure><p>Instruct students to verbally describe the quadratic parent function.</p><p>Students should be able to describe how the quadratic parent function y = x² squares each input (<i>x</i>) and returns the output (<i>y</i>). The <i>rate of change</i> decreases before increasing faster than a linear function. The <i>vertex</i> is at the origin (0, 0), and the <i>domain</i> is all real numbers (-∞, ∞), with a <i>range</i> of all real numbers greater than or equal to 0, or [0, ∞). Thus, this graph has a <i>minimum</i>, which is the vertex, or (0, 0).</p><p>Tossing a ball, shooting an arrow, and other projectile-related challenges, as well as estimating profits, losses, or costs, are some examples.</p><p>Example problem: Kevin throws a penny off of his hotel balcony onto the sidewalk below. The balcony is 45 feet above the ground. The function <i>h</i> = 36<i>t</i>² + 45 calculates the height of the coin after <i>t</i> seconds.</p><p>Make a table of values, graph the function, and describe the overall graph, including the vertex, domain, and range.</p><figure class="image"><img style="aspect-ratio:252/186;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_150.png" width="252" height="186"></figure><figure class="image"><img style="aspect-ratio:431/451;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_151.png" width="431" height="451"></figure><p>Remind students that the graph depicts the behavior of the measurements of the coin (distance and time) as it falls, rather than the coin's path.</p><p>This graph raises various questions that need to be answered:</p><ul><li>What happens at 0 feet?</li><li>What does the negative <i>h</i>-value indicate?</li><li>When does the coin touch the sidewalk? How do we figure out the answer?</li><li>What is the domain and range? What is a reasonable domain and range given the context of the problem?</li><li>What is a vertex, and what does it mean? What's the orientation?</li></ul><p>Let's answer each question.</p><ul><li>What happens at 0 feet?&nbsp;</li></ul><p><i>At 0 feet, the coin hits the sidewalk.</i></p><ul><li>What does a negative point indicate?&nbsp;</li></ul><p><i>The negative point indicates that, after 2 seconds, the coin is 99 feet below the ground. This point is not reasonable in the context of the problem.</i></p><ul><li>When does the coin touch the sidewalk? How do we figure out the answer?</li></ul><p><i>To figure out when the coin touches the sidewalk, we must set the function equal to 0. In other words, we'll replace h with 0. So,</i></p><figure class="image"><img style="aspect-ratio:487/213;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_152.png" width="487" height="213"></figure><p><i>After 1.12 seconds, the coin reaches the sidewalk.</i></p><ul><li>What is the domain and range? What is a reasonable domain and range given the context of the problem?</li></ul><p><i>The domain is [0,2). The range is (-∞, 45]. However, the reasonable domain and range are very different. The reasonable domain is [0,1.12]. The reasonable range is [0, 45]. (\(\sqrt{5} \over 2\)) gives an exact solution.</i></p><p><i>In other words, you only have seconds from 0 to 1.12 until the coin hits the ground. The elevation ranges from 45 feet to 0 feet above earth, or ground level.</i></p><ul><li>What is a vertex, and what does it represent? What's the orientation?</li></ul><p><i>The vertex of a parabola is either its highest or lowest point. In this situation, the vertex is at (0, 45). It is a maximum, which means the parabola opens downward. It is the moment at which the coin reaches its maximum height before dropping. The graph's orientation is negative since it opens downward. The coin is dropped, and the height lowers over time.</i></p><p><strong>"What else does the graph reveal?"</strong></p><p>We can conclude that there is no constant rate of change, which means that the coin does not move at the same rate from his hand to the sidewalk.</p><p><strong>"How do we know that this type of scenario is best represented by a quadratic function or parabola? Why not a linear or absolute value function? Why not an exponential function?"</strong> Note that students may lack experience with these functions.</p><p>Place each student with a partner and work together to develop a real-world quadratic 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 characterize them in everyday language?</p><p><strong>"Let's take a look at some quadratic function transformations. When I mention 'transformations,' what comes to mind? What examples can you provide? When would we ever need to learn about transformations or apply them to solve a problem?"</strong></p><p>Review the quadratic parent function and introduce students to several altered quadratic functions. A parent function is a function that has not been transformed.</p><p><strong>"Let's make a list of possible transformed quadratic functions. Imagine you're changing the quadratic parent function. How could you do that? What is the number of ways? Let's make some of them."</strong></p><p>Here are some examples:</p><p>y = 4x²</p><p>y = 2x²</p><p>y = –4x²</p><p>y = –2x²</p><p>y = –x²</p><p>y = x² + 4</p><p>y = (x + 4)²</p><p>y = x² – 4</p><p>y = (x – 4)²</p><p>y = (x + 4)² – 4</p><p>y = –4(x – 4)²</p><p><br><strong>"Now let's examine each kind of transformation, starting with simple transformations and ending with combinations of shifts."</strong></p><p>Have students guess how each transformation will look. Divide the students into groups of three or four. Ask students to fill out the chart below. Students will draw the predicted graph before creating the table of numbers and building the actual graph. Students will compare the transformed functions to the quadratic parent function. Use one or more of the examples provided above. Ask students to look into other functions by adding more rows to the table.</p><p>&nbsp;</p><p>After students have graphed each function by hand, have them confirm their results with a graphing calculator or GeoGebra. Students can use the graphing calculator's table feature or enter the function's equation into the <i>y</i> = screen to graph it. GeoGebra will let students plot points and graph the function.</p><p>Students can have a pleasant conversation about the basis for their predictions. Students will uncover the rationale behind their thinking about reducing and stretching a graph. Were their intuitions correct? <strong>"How can we comprehend how the variables in the equation of a stretched/compressed graph relate to its actual appearance? In other words, how can we comprehend the relationship without memorizing a rule?"</strong></p><p><strong>"What did we learn about shifts inside and outside of parentheses? Were the outcomes what you expected? Why, or why not? What have you learned? How did you make the connections, thereby expanding your conceptual understanding?"</strong></p><p><strong>"What occurred when we combined transformations with quadratic functions? Which transformation did you try first? Did the order matter? Why, or why not?"</strong></p><p>Hold the discussion first, then conduct a more in-depth analysis of each piece. This allows students to express their ideas/concerns first, establishing a foundation from which to catapult.</p><p>Distribute copies of the Lesson 2 Graphic Organizer (M-A2-7-2_Lesson 2 Graphic Organizer) to practice recognizing the features of quadratic functions.</p><p><strong>"Compare the equations, tables, and graphs for each of our instances. In addition, we will provide a written summary of what occurred. We shall note the domain, range, and vertex beneath each graph."</strong></p><p><strong>Transformations of the Quadratic Parent Function </strong><i><strong>y = x²</strong></i></p><figure class="image"><img style="aspect-ratio:623/722;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_154.png" width="623" height="722"></figure><figure class="image"><img style="aspect-ratio:623/674;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_155.png" width="623" height="674"></figure><figure class="image"><img style="aspect-ratio:622/676;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_156.png" width="622" height="676"></figure><figure class="image"><img style="aspect-ratio:620/809;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_157.png" width="620" height="809"></figure><figure class="image"><img style="aspect-ratio:623/811;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_158.png" width="623" height="811"></figure><figure class="image"><img style="aspect-ratio:620/297;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_159.png" width="620" height="297"></figure><p><strong>"Now, consider all functions graphed on the same graph, including the parent quadratic function. We shall assign a letter to each function."</strong> See the chart above.</p><figure class="image"><img style="aspect-ratio:530/426;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_160.png" width="530" height="426"></figure><p><br><strong>Note: The graphs for A, B, C, D, E, F, G, H, I, J, and K are dark blue, green, red, light blue, purple, yellow, light green, blue-green, pink, purple, and green, respectively.</strong></p><p>Divide the students into pairs. Have each group decide the domain, range, and vertex for the changed functions listed above. The domain and range should be stated using both interval notation and word form.</p><p><strong>Part 2</strong></p><p>Students can experiment with quadratic functions and function transformations using NLVM's virtual Grapher applet, which is available at <a href="http://nlvm.usu.edu/en/nav/frames_asid_109_g_4_t_2.htmlopen=activities&amp;from=category_g_4_t_2.html">http://nlvm.usu.edu/en/nav/frames_asid_109_g_4_t_2.htmlopen=activities&amp;from=category_g_4_t_2.html</a></p><figure class="image"><img style="aspect-ratio:309/218;" src="https://storage.googleapis.com/worksheetzone/images/Screenshot_161.png" width="309" height="218"></figure><p><br>Students can use the applet to investigate the parent function y = x² and transformations, such as reflection across the x-axis, y = -x². Students should also practice various transformations, such as right and left slides, as well as up and down shifts. Students should take notes and make observations on the relationships between the function equation and the output 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 how the width of the function varies as the value of the coefficient a changes.</p><p>Share the Quadratic Modeling Worksheet (M-A2-7-2_Quadratic_Modeling_Worksheet and KEY).</p><p><strong>Review</strong></p><p>1. Conduct a matching activity where students match graphs and tables of different quadratic functions to their equations. Include 5–10 graphs and tables.</p><p>2. Have students create rules for transforming quadratic functions. Algorithms should be thorough and specific.</p><p>3. Ask students to create another word problem using a quadratic function. Students should approach the problem with a variety of representations.</p><p><strong>Extension:</strong></p><p>• Encourage students to create and represent quadratic functions of the form <i>y = ax² + bx + c</i>. Ask them to come up with a real-world problem that uses this type of function. Students should explain the meanings of <i>a</i>, <i>b</i>, and <i>c</i> in the context of the problem.</p>