Lesson 20: Four Interesting Transformations of Functions

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Mae Norman
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1 Student Outcomes Students apply their understanding of transformations of functions and their graphs to piecewise functions. Lesson Notes In Lessons students study translations and scalings of functions and their graphs. In, these transformations are applied in combination to piecewise functions. Students should become comfortable visualizing how the graph of a transformed piecewise function will relate to the graph of the original piecewise function. Classwork Opening Exercise (6 minutes) Have students work individually or in pairs to complete the Opening Exercise. This exercise highlights MP.7 since it calls on students to interpret the meaning of in the context of a graph. Opening Exercise Fill in the blanks of the table with the appropriate heading or descriptive information. Graph of Vertical Horizontal Translate Translate up by units Translate down by units Translate right by units Translate left by units Vertical stretch by a factor of Horizontal stretch by a factor of Scale by scale factor Vertical shrink by a factor of Vertical shrink by a factor of and reflection over axis Horizontal shrink by a factor of Horizontal shrink by a factor of and reflection across axis Vertical stretch by a factor of and reflection over axis Horizontal stretch by a factor of and reflection over axis Date: 4/7/14 256

2 In Lesson 15, we discovered how the absolute value function can be written as a piecewise function. Example 1 and the associated exercises are intended to help students reexamine how piecewise functions behave. Example 1 (3 minutes) Example 1 A transformation of the absolute value function,, is rewritten here as a piecewise function. Describe in words how to graph this piecewise function.,, First, I would graph the line for values less than, and then I would graph the line for values greater than or equal to. Exercises 1 2 (15 minutes) Exercises Describe how to graph the following piecewise function. Then graph below.,.,, The function can be graphed of as the line for values less than or equal to, the graph of the line. for values greater than and less than, and the graph of the line for values greater than or equal to. Date: 4/7/14 257

3 2. Using the graph of below, write a formula for as a piecewise function...,,,, or..,,, Example 2 (10 minutes) Students translate and scale the graph of a piecewise function. Example 2 The graph of a piecewise function is shown. The domain of is, and the range is. a. Mark and identify four strategic points helpful in sketching the graph of.,,,,,, and, Date: 4/7/14 258

4 b. Sketch the graph of and state the domain and range of the transformed function. How can you use part (a) to help sketch the graph of? Domain:, range:. For every point, in the graph of, there is a point, on the graph of. The four strategic points can be used to determine the line segments in the graph of by graphing points with the same original coordinate and times the original coordinate (,,,,,, and, ). c. A horizontal scaling with scale factor of the graph of is the graph of. Sketch the graph of and state the domain and range. How can you use the points identified in part (a) to help sketch? Domain:.., range:. For every point, in the graph of, there is a point, on the graph of. The four strategic points can be used to determine the line segments in the graph of by graphing points with one half the original coordinate and the original coordinate (.,,.,,.,, and., ). Date: 4/7/14 259

5 Exercises 3 4 (5 minutes) Exercises How does the range of in Example 2 compare to the range of a transformed function, where, when? For every point, in the graph of, there is a point, in the graph of, where the number is a multiple of each. For values of, is a vertical scaling that appears to stretch the graph of. The original range, for becomes for the function. 4. How does the domain of in Example 2 compare to the domain of a transformed function, where, when? (Hint: How does a graph shrink when it is horizontally scaled by a factor?) For every point, in the graph of, there is a point, in the graph of. For values of, is a horizontal scaling by a factor that appears to shrink the graph of. This means the original domain, for becomes for the function. Closing (2 minutes) The transformations that translate and scale familiar functions, like the absolute value function, also apply to piecewise functions and to any function in general. By focusing on strategic points in the graph of a piecewise function, we can translate and scale the entire graph of the function by manipulating the coordinates of those few points. Exit Ticket (4 minutes) Date: 4/7/14 260

6 Name Date Exit Ticket The graph of a piecewise function is shown below. Let 2, 1 2 2, and Graph,, and on the same set of axes as the graph of. Date: 4/7/14 261

7 Exit Ticket Sample Solutions The graph of a piecewise function is shown below. Let,, and. Graph,, and on the same set of axes as the graph of. Date: 4/7/14 262

8 Problem Set Sample Solutions 1. Suppose the graph of is given. Write an equation for each of the following graphs after the graph of has been transformed as described. Note that the transformations are not cumulative. a. Translate units upward. b. Translate units downward. c. Translate units right. d. Translate units left. e. Reflect about the axis. f. Reflect about the axis. g. Stretch vertically by a factor of. h. Shrink vertically by a factor of. i. Shrink horizontally by a factor of. j. Stretch horizontally by a factor of. 2. Explain how the graphs of the equations below are related to the graph of. a. The graph is a vertical stretch of by a factor of. Date: 4/7/14 263

9 b. The graph of is translated right units. c. The graph is a vertical stretch of by a factor of and reflected about the axis. d. The graph is a horizontal shrink of by a factor of. e. The graph is a vertical stretch of by a factor of and translated down units. 3. The graph of the equation is provided below. For each of the following transformations of the graph, write a formula (in terms of ) for the function that is represented by the transformation of the graph of. Then draw the transformed graph of the function on the same set of axes as the graph of. a. A translation units left and units up. Date: 4/7/14 264

10 b. A vertical stretch by a scale factor of. c. A horizontal shrink by a scale factor of. 4. Reexamine your work on Example 2 and Exercises 3 and 4 from this lesson. Parts (b) and (c) of Example 2 asked how the equations and could be graphed with the help of the strategic points found in (a). In this problem, we investigate whether it is possible to determine the graphs of and by working with the piecewise linear function directly. a. Write the function in Example 2 as a piecewise linear function...,,, Date: 4/7/14 265

11 b. Let. Use the graph you sketched in Example 2, part (b) of to write the formula for the function as a piecewise linear function.,,, c. Let. Use the graph you sketched in Example 2, part (c) of to write the formula for the function as a piecewise linear function..,,, d. Compare the piecewise linear functions and to the piecewise linear function. Did the expressions defining each piece change? If so, how? Did the domains of each piece change? If so how? Function : Each piece of the formula for is times the corresponding piece of the formula for. The domains are the same. Function : Each piece of the formula for is found by substituting in for in the corresponding piece of the formula for. The length of each interval in the domain of is the length of the corresponding interval in the domain of. Date: 4/7/14 266

Lesson 19: Four Interesting Transformations of Functions

Lesson 19: Four Interesting Transformations of Functions Student Outcomes Students examine that a horizontal scaling with scale factor of the graph of corresponds to changing the equation from to 1. Lesson Notes In this lesson, students study the effect a horizontal