5.4 Take Some Time to Reﬂect Reﬂections of Linear and Exponential Functions LEARNING GOALS KEY TERMS In this lesson, you will: t Reflect linear and exponential t reflection t line of reflection functions vertically. t Reflect linear and exponential functions horizontally. t Determine characteristics of graphs after transformations. Y ou are already familiar with many different types of “reflections” in mathematics. When a negative sign is present, this is a good indication of a reflection. For example, 25 is a “reflection” of 5 over 0 on the number line. 2 1 The power 21 5 __, but 221 5 __. 1 2 In this lesson, you will learn about reflecting functions. Watch out for those negative signs! © 2012 Carnegie Learning 5 5.4 Reflections of Linear and Exponential Functions 327 Problem 1 Basic linear and exponential functions are used to explore vertical and horizontal reflections. Students will write given functions in terms of basic functions, describe the operation performed on the basic function, and then a graphing calculator to compare the graph of the basic function to the graph of the reflected functions. Students then complete a table of values comparing the ordered pairs of the graphs of the basic function to the ordered pairs of the graphs of the reflected functions. They will generalize about the constraints involved when linear or exponential functions are reflected and are able to recognize the graphical implications of the movements using only the equations of the functions. PROBLEM 1 Reﬂections Consider the three exponential functions shown, where h(x) 5 2x is the basic function. r h(x) 5 2x r m(x) 5 2(2x) r n(x) 5 2(2x) 1. Write the functions m(x) and n(x) in terms of the basic function h(x). 2h(x) m(x) 5 n(x) 5 h(2x) 2. Compare m(x) to h(x). Does an operation performed on h(x) or on the argument of h(x) result in the equation for m(x)? What is the operation? The function h(x) is multiplied by 21 to result in the equation for m(x). 3. Compare n(x) to h(x). Does an operation performed on h(x) or on the argument of h(x) result in the equation for n(x)? What is the operation? The argument of the function h(x) is multiplied by 21 to result in the equation for n(x). 4. Use a graphing calculator to graph each function with the bounds [210, 10] 3 [210, 10]. Then, sketch the graph of each function. Label each graph. y h(x) x 5 2x n(x) x 5 Before you press the GRAPH key, make a prediction about the shapes of m(x) and n(x). x 2(2x) x m(x) x 5 2(2x) Grouping 5 Guiding Questions for Share Phase, Questions 1 through 4 t Is there a difference between the function 2h(x) and the function h(2x)? What is the difference? t Does the function 2h(x) have an asymptote? Where? t Does the function h(2x) have t Is there a difference between the function m(x) and the function n(x)? What is the difference? t Does the function m(x) have an asymptote? Where? t Does the function n(x) have an asymptote? Where? t Which function is reflected over the y-axis? t Which function is reflected over the x-axis? an asymptote? Where? 328 Chapter 5 Exponential Functions © 2012 Carnegie Learning Have students complete Questions 1 through 7 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 5 through 7 t Looking at only the equation of an exponential function, how can you determine if the graph of the reflected function will be reflected over the y-axis? t Looking at only the equation of an exponential function, how can you determine if the graph of the reflected function will be reflected over the x-axis? 5. Compare the graphs of m(x) and n(x) to the graph of the basic function h(x). What do you notice? The graph of m(x) is a mirror image of the graph of h(x), reﬂected over the horizontal line y 5 0. The graph of n(x) is a mirror image of the graph of h(x), reﬂected over the vertical line x 5 0. 6. Complete the table of ordered pairs for the three given functions. h(x) 5 2x m(x) 5 2(2x) __1 1) (22, __ 4 (22, 22, 2, 1) (21, __ 2 21, 1, (21, 2 4 ) __1 2 2 ) n(x) 5 2(2x) ( 2 1) , __ 4 ( 1 1) , __ 2 , 1) (0, 1) (0, 21 ) ( 0 (1, 2) (1, 22 ) ( 21 , 2) (2, 4) (2, 24 ) ( 22 , 4) Grouping Ask a student to read the information and definition following Question 7. Discuss as a class. 7. Use the table to compare the ordered pairs of the graphs of m(x) and n(x) to the ordered pairs of the graph of the basic function h(x). What do you notice? For the same x-coordinate, the y-coordinate of m(x) is the opposite of the y-coordinate of h(x). For the same y-coordinate, the x-coordinate of n(x) is the opposite of the x-coordinate of h(x). A reflection of a graph is a mirror image of the graph about a line of reflection. A line of reflection is the line that the graph is reflected about. A horizontal line of reflection affects the y-coordinates, and a vertical line of reflection affects the x-coordinates. You can use the coordinate notation shown to indicate a reflection about a horizontal line of reflection. When the negative is on the outside of the function, like –g(x), all the y-values become the opposite of the y-values of g(x). The x-values remain unchanged. 5 © 2012 Carnegie Learning (x, y) → (x, 2y) 5.4 Reflections of Linear and Exponential Functions 329 Grouping t Have students complete You can also use the coordinate notation shown to indicate a reflection about a vertical line of reflection. Questions 8 and 9 with a partner. Then share the responses as a class. When the negative is on the inside of the function, like g(–x), all the x-values become the opposite of the x-values of g(x). The y-values remain unchanged. (x, y) → (2x, y) t Have students complete 8. Which function represents a reflection of h(x) over a horizontal line? Which function represents a reflection of h(x) over a vertical line? Questions 10 and 11 with a partner. Then share the responses as a class. The function m(x) is a reﬂection of h(x) over a horizontal line. The function n(x) is a reﬂection of h(x) over a vertical line. 9. Describe each graph in relation to its basic function. Guiding Questions for Share Phase, Questions 8 and 9 t When coordinate notation a. Compare f(x) 5 2(bx) to the basic function h(x) 5 bx. The graph of f(x) is a reﬂection of the graph of h(x) over the horizontal line y 5 0. is used to represent a horizontal reflection, is the negation associated with the x-coordinate or is the negation associated with the y-coordinate? b. Compare f(x) 5 b(2x) to the basic function h(x) 5 bx. The graph of f(x) is a reﬂection of the graph of h(x) over the vertical line x 5 0. 10. The graph of a function w(x) is shown. Sketch the graphs of w9(x) and w0(x). a. w9(x) 5 2w(x) b. w0(x) 5 w(2x) t Does a vertical reflection y affect the x-coordinate or the y-coordinate of the basic function? 4 3 2 1 w(x) w(2x) Guiding Questions for Share Phase, Questions 10 and 11 t Is there a difference between t Do you need to create a table of values to graph the reflections? Why or why not? 330 4 x 2w(x) 23 24 11. Write the equation of each function after a reflection about the horizontal line y 5 0. Then, write the equation after a reflection about the vertical line x 5 0. a. a(x) 5 5x Reflection about y 5 0: a9(x) 5 over the y-axis? describing the function w(x) to graph the reflections? Why or why not? 3 22 t Which function is reflected t Do you need the equation 2 21 the function 2w(x) and the function w(2x)? What is the difference? t Which function is reflected over the x-axis? 1 Reflection about x 5 0: a0(x) 5 25 x 2x 5 t What points did you use to graphically reflect the function w(x)? t How does the graph of a(x) 5 5x compare to the graph of a(x) 5 25x? t How does the graph of a(x) 5 5x compare to the graph of a(x) 5 52x? t How does the graph of b(x) 5 22x2 compare to the graph of b(x) 5 2(2x)2? t How does the graph of c(x) 5 __45x3 compare to the graph of c(x) 5 2__45 x3? t How does the graph of c(x) 5 __45 x3 compare to the graph of c(x) 5 __45 (2x)3 ? Chapter 5 Exponential Functions © 2012 Carnegie Learning 5 0 24 23 22 21 t A horizontal reflection of a function involves negating what part of the basic function? b. b(x) 5 22x2 t A vertical reflection of a function involves negating what part of the basic function? Reflection about y 5 0: b9(x) 5 2x2 Reflection about x 5 0: b0(x) 5 22x2 5x3 c. c(x) 5 __ 4 Reflection about y 5 0: c9(x) 5 __5 2 x 4 __5(2x) 3 Reflection about x 5 0: c0(x) 5 Problem 2 A horizontal reflection, s(x) 5 (2x) and a vertical reflection, r(x) 5 2(x) are performed on the basic function g(x) 5 x resulting in the same expression and graphically, the same line. Students will complete a table of values used to compare the ordered pairs of each function to conclude that the reflection of the basic function is the same over the x- and y-axes. 3 PROBLEM 2 4 Linear Functions . . . Another Curious Case! Consider the three linear functions shown, where g(x) 5 x is the basic function. r g(x) 5 x r r(x) 5 2(x) r s(x) 5 (2x) 1. Write the functions r (x) and s(x) in terms of the basic function g(x). 2g(x) r(x) 5 s(x) 5 g(2x) 2. Use a graphing calculator to graph each function with the bounds [210, 10] 3 [210, 10]. Then, sketch the graph of each function. Label each graph. y r( x) 5 2(x) and g(x) g( ( xx)) 5 x s(x) x 5 (2x) x Grouping x © 2012 Carnegie Learning Have students complete Questions 1 through 6 with a partner. Then share the responses as a class. Guiding Questions for Share Phase, Questions 1 through 6 t Which function represents 5 3. Compare the graphs of r(x) and s(x) to the graph of the basic function g(x). What do you notice? The graphs of r(x) and s(x) are the same. They are both mirror images of the graph of g(x). the vertical reflection of g(x)? t Which function represents the horizontal reflection of g(x)? t What is the argument of g(x)? t If the functions simplify to t Can you think of any other basic function in which the graph of the horizontal and the graph of the vertical reflection would result in the same line? the same expression, what graphical implications might this have? 5.4 Reflections of Linear and Exponential Functions 331 4. Complete the table of ordered pairs for the three given functions. g(x) 5 x r(x) 5 2(x) s(x) 5 (2x) (22, 22) (22, 22, 2, 2 ) ( 2 , 22) (21, 21) (21, 21, 1, 1 ) ( 1 , 21) (0, 0) (0, (1, 1) (1, 21 ) ( 21 , 1) (2, 2) (2, 22 ) ( 22 , 2) 0 ) ( 0 , 0) 5. Use the table to compare the ordered pairs of the graphs of r(x) and s(x) to the ordered pairs of the graph of the basic function g(x). What do you notice? For the same x-coordinate, the y-coordinate of r(x) is the opposite of the y-coordinate of g(x). For the same y-coordinate, the x-coordinate of s(x) is the opposite of the x-coordinate of g(x). 6. Which function represents a reflection of the basic function g(x) over a vertical line? Which function represents a reflection of the basic function g(x) over a horizontal line? Why do you think they produce the same graph? The function r(x) is a reﬂection of the basic function g(x) over a horizontal line. The function s(x) is a reﬂection of the basic function g(x) over a vertical line. They produce the same graph because the reﬂection of the basic function g(x) 5 x is the same over the y- and x-axes. © 2012 Carnegie Learning 5 332 Chapter 5 Exponential Functions Problem 3 Students are given several descriptions of functions and will write an equation and sketch a graph of the function. A Who’s Correct? question provides two different methods of performing transformations that result in the same graph. In the last activity, students will name all possible transformations performed on a basic function that would produce a graph of a described function. PROBLEM 3 Characteristics of Graphs after Transformations 1. Use the given characteristics to write an equation and then sketch a graph of g(x). a. Write an equation and sketch a graph that: r is an exponential function, r is continuous, r is increasing, and r is translated 2 units to the left of f(x) 5 2x. x12 2 Equation: g(x) 5 y x Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. b. Write an equation and sketch a graph that: r is an exponential function, r is continuous, Guiding Questions for Share Phase, Question 1 t Do all exponential functions r is increasing, and r is translated 5 units down from f(x) 5 2x. Equation: g(x) 5 have an asymptote? Why or why not? x 2 25 y t If the function is exponential, what is the basic function? 5 x © 2012 Carnegie Learning t What feature of the exponential function ensures that the function is increasing? t What feature of the exponential function ensures that the function has been translated 2 units to the left? t What feature of the exponential function ensures that the function has been translated 5 units down? t What feature of the exponential function ensures that the function is decreasing? t What feature of the exponential function ensures that the function has been reflected over the y-axis? 5.4 Reflections of Linear and Exponential Functions 333 Guiding Questions for Share Phase, Question 2 t What is the basic function Jacob and Kate are using? t Who used a vertical reflection? t Who used a horizontal reflection? c. Write an equation and sketch a graph that: r is an exponential function, r is continuous, r is decreasing, and r is a reflection of f(x) 5 2x over the line x 5 0. 22x Equation: g(x) 5 y t Who used a upward translation? t Who used a downward x translation? t Could Jacob and Kate both be correct? ? 2. Jacob and Kate are comparing the two graphs shown. y y f(x) x x g(x) Jacob says that to get the graph of g(x), first translate f(x) down 3 units, and then reflect over the line y 5 0. Kate says that to get the graph of f(x), first reflect g(x) over the line y 5 0, and then translate up 3 units. Who is correct? Explain your reasoning. Both Jacob and Kate are correct. To transform f(x) to g(x), you can translate and then reﬂect. To transform g(x) to f(x), you can reﬂect and then translate. © 2012 Carnegie Learning 5 334 Chapter 5 Exponential Functions Guiding Questions for Share Phase, Question 3 t If the graph of g(x) and the graph of f(x) have a different y-intercept, what does this imply about the transformation that was used? 3. Choose the transformations in the box performed on f(x) that would produce the graph of g(x). Then sketch the graph of g(x) on the coordinate plane. vertical translation horizontal translation reflection over the line y 5 0 reflection over the line x 5 0 a. The graph of f(x) is shown. y t If the graph of g(x) and the graph of f(x) have a different y-intercept, what does this imply about the transformation that was not used? t If the graph of g(x) and the graph of f(x) have the same asymptote, what does this imply about the transformation that was used? x The graph of g(x) has a different y-intercept than the graph of f(x), but the same asymptote as f(x). Possible transformation(s) on f(x) to produce g(x): Sketch of g(x) y Sample answers. horizontal translation reﬂection over the line y 5 0 x t If the graph of g(x) and the graph of f(x) have the same asymptote, what does this imply about the transformation that was not used? b. The graph of f(x) is shown. y x t If the graph of g(x) and © 2012 Carnegie Learning the graph of f(x) have the same y-intercept, what does this imply about the transformation that was used? The graph of g(x) has the same y-intercept as f(x), but g(x) is decreasing. Possible transformation(s) on f(x) to produce g(x): Sketch of g(x) y t If the graph of g(x) and the graph of f(x) have the same y-intercept, what does this imply about the transformation that was not used? t If the graph of g(x) is decreasing and the graph of f(x) is increasing, what does this imply about the transformation that was used? 5 reﬂection over the line x 5 0 x t If the graph of g(x) is decreasing and the graph of f(x) is increasing, what does this imply about the transformation that was not used? t If the graph of g(x) has a different asymptote than the graph of f(x), what does this imply about the transformation that was used? t If the graph of g(x) has a different asymptote than the graph of f(x), what does this imply about the transformation that was used? 5.4 Reflections of Linear and Exponential Functions 335 c. The graph of f(x) is shown. y x The graph of g(x) has a different asymptote than the graph of f(x), and g(x) is increasing. Possible transformation(s) on f(x) to produce g(x): Sketch of g(x) y vertical translation x Be prepared to share your solutions and methods. 336 Chapter 5 Exponential Functions © 2012 Carnegie Learning 5

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