The Graph Scale-Change Theorem

Transcription

1 Lesson 3-5 Lesson 3-5 The Graph Scale-Change Theorem Vocabular horizontal and vertical scale change, scale factor size change BIG IDEA The graph of a function can be scaled horizontall, verticall, or in both directions at the same time. Vertical Scale Changes Consider the graph of = f1() = shown both in the graph and function table below. What happens when ou multipl all the -values of the graph b? What would the resulting graph and table look like? Activit 1 will help ou answer these questions. Mental Math What is R 70 (1, 0)? Activit 1 MATERIALS Graphing utilit or slider graph application from our teacher Step 1 Graph f1() = with window and Step Graph f() = 3( ) = 3 f1() on the same aes. Fill in the table of values for f1() and f() onl. Describe how the f() values relate to the f1() values. f1() f() 0.5 f1() 1.5 f1() f1()?????????? 7????? 10????? Step 3 Repeat Step with f3() = b( ) and use a slider to var the value of b. Set the slider to 0.5, 1.5, and then. For each of these b-values, use a function table to fi ll in a column of the table in Step. Describe how the f3() values relate to the corresponding values of f1(). In Step of Activit 1, we sa that the graph of f is the image of the graph of f1 under a vertical scale change of magnitude 3. Each point on the graph of f is the image of a point on the graph of f under the mapping (, ) (, 3). You can create the same change b replacing with _ 3 in the equation for f1, because _ 3 = is equivalent to = 3( ), or = 3f1() in function notation. Similarl, if ou replace with in the original equation, ou obtain = , which is equivalent to = 1_ ( ) or = 1_ f(). The Graph Scale-Change Theorem 179

2 Chapter 3 QY Horizontal Scale Changes Replacing the variable b k in an equation results in a vertical scale change. What happens when the variable is replaced b _? B? Activit QY Replacing b _ in the equation for f1 ields an equation equivalent to =?. What is the effect on the graph of f? MATERIALS Graphing utilit or slider graph application provided b our teacher. Step 1 Consider the graph of f1 in Activit 1. Complete the table at the right. f1()? 1?? Step If f() = f1 ( _ a ), then f() = ( _ a ) ( _ a ) - ( _ a ). Graph f and use a slider to var the value of a. Step 3 a. What are the - and -intercepts of f1? b. How do the intercepts of f change as a changes? The graph of f in Activit is the image of the graph of f1 under a horizontal scale change of magnitude a. Each point on the graph of f is the image of a point on the graph of f1 under the mapping (, ) (a, ). Replacing b _ in the equation doubles the -values of the preimage points while the corresponding -values remain the same. Accordingl, the -intercepts of the image are two times as far from the -ais as the -intercepts of the preimage. The Graph Scale-Change Theorem In general, a scale change centered at the origin with horizontal scale factor a 0 and vertical scale factor b 0 is a transformation that maps (, ) to (a, b). The scale change S can be described b S: (, ) (a, b) or S(, ) = (a, b). If a = 1 and b 1, then the scale change is a vertical scale change. If b = 1 and a 1, then the scale change is a horizontal scale change. When a = b, the scale change is called a size change. Notice that in the preceding instances, replacing b _ in an equation for a function results in the scale change S: (, ) (, ); and replacing b _ leads to the 3 scale change S: (, ) (, 3). These results generalize. 10 Transformations of Graphs and Data

3 Lesson 3-5 Graph Scale-Change Theorem Given a preimage graph described b a sentence in and, the following two processes ield the same image graph: (1) replacing b _ a and b _ in the sentence; b () appling the scale change (, ) (a, b) to the preimage graph. Proof Name the image point (, ). So = a and = b. Solving for and gives _ a = and _ b =. The image of = f() will be _ a = f ( _ a ). The image equation is written without the primes. Unlike translations, scale changes do not produce congruent images unless a = b = 1. Notice also that multiplication in the scale change corresponds to division in the equation of the image. This is analogous to the Graph-Translation Theorem in Lesson 3-, where addition in the translation (, ) ( + h, + k) corresponds to subtraction in the image equation - k = f( - h). GUIDED Eample 1 The relation described b + = 5 is graphed at the right. a. Find images of points labeled A F on the graph under S: (, ) (, ). b. Cop the circle onto graph paper; then graph the image on the same aes. c. Write an equation for the image relation. Solution a. Cop and complete the table below. b. Plot the preimage and image points on graph paper and draw a smooth curve connecting the image points. A partial graph is drawn below. c. According to the Graph Scale-Change Theorem, an equation for an image under S: (, ) (, ) can be found b replacing b? in the equation for the preimage. The result is the equation?. D (0, 5) E ( -3, ) C (3, ) B (, 3) A (5, 0) F (, -3) Preimage Image Point A B???? C???? D???? E???? F???? B (, 3) B' (, 3) A (5, 0) A' (10, 0) 5 10 The Graph Scale-Change Theorem 11

4 Chapter 3 Eample A graph and table for = f() are given at the right. Draw the graph of _ 3 = f(). Solution Rewrite _ 3 = f() as _ 3 = f ( _ 1_. B the Graph Scale-Change ) Theorem, replacing b _ 1_ and b _ 3 is the same as appling the scale change (, ) ( 1_, 3 ). So, A = (, ) ( 3, ) = A and B = ( 3, 1) ( 3/, 3) = B C = (0, 1) (0, 3) = C D = (, 3) (1, 9) = D E = (, 0) (3, 0) = E. The graph of the image is shown at the right. Negative Scale Factors Notice what happens when a scale factor is 1. Consider the horizontal and vertical scale changes H and V with scale factors equal to 1. H: (, ) (, ) and V: (, ) (, ) In H, each -value is replaced b its opposite, which produces a reflection over the -ais. Similarl, in V, replacing b produces a reflection over the -ais. More generall, a scale factor of k combines the effect of a scale factor of k and a reflection over the appropriate ais. Questions COVERING THE IDEAS f() True or False Under ever scale change, the preimage and image are congruent.. Under a scale change with horizontal scale factor a and vertical factor b, the image of (, ) is?. 3. Refer to the Graph Scale-Change Theorem. Wh are the restrictions a 0 and b 0 necessar?. If S maps each point (, ) to ( _, ), give an equation for the image of = f() under S. A - = f() - B A' C - D E E' B' C' D' = f() 3 1 Transformations of Graphs and Data

5 Lesson Consider the function f1 used in Activities 1 and. a. Write a formula for f1 ( 3) _. b. How is the graph of = f1 ( 3) _ related to the graph of = f1()?. Multiple Choice Which of these transformations is a size change? A (, ) (3, 3) B (, ) (3, ) C (, ) ( + 3, + 3) D (, ) ( _ 3, ) 7. Functions f and g with f() = and g() = 1_ f() are graphed at the right. a. What scale change maps the graph of f to the graph of g? b. The -intercepts of f are at = 1_ and = 3. Where are the -intercepts of g? c. How do the -intercepts of f and g compare? - d. The verte of the graph of f is (1.5,.15). What is the verte of the graph of g?. Consider the parabola with equation =. Let S(, ) = (, _ 7). a. Find the images of ( 3, 9), (0, 0), and ( 1_, 1_ under S. ) b. Write an equation for the image of the parabola under S. 9. The graph of a function f is shown at the right. a. Graph the image of f under S(, ) = ( 1_, 3 ). b. Find the - and -intercepts of the image. c. Find the coordinates of the point where the -value of the image of f reaches its maimum. 10. Give another name for the horizontal scale change of magnitude Describe the scale change that maps the graph of = onto the graph of = _ 1. APPLYING THE MATHEMATICS 1. Refer to the parabolas at the right. The graph of g is the image of the graph of f under what a. horizontal scale change? b. vertical scale change? c. size change? 13. Write an equation for the image of the graph of = + 1_ under each transformation. a. S(, ) = (, ) b. S(, ) = ( _ 3, ) 1. A scale change maps (10, 0) onto (, 0) and ( 5, ) onto ( 1, ). What is the equation of the image of the graph of f() = 3 - under the scale change? f() = g() = f() f() = 10 g() = (-, ) (, ) (-, ) (, ) (-1, 1) (1, 1) (, 1) (, 1) The Graph Scale-Change Theorem 13

6 Chapter 3 In 15 and 1, give a rule for a scale change that maps the graph of f onto the graph of g g() f() g() f() REVIEW In 17 and 1, an equation for a function is given. Is the function odd, even, or neither? If the function is odd or even, prove it. (Lesson 3) 17. f() = (5 + ) 3 1. g() = If f() = g() for all in the common domain of f and g, how are the graphs of f and g related? (Lesson 3) 0. One of the parent functions presented in Lesson 3-1 has a graph that is not smmetric to the -ais, -ais, or origin. It has the asmptote = 0. Which is it? (Lesson 3-1) The table at the right shows the number of injuries on different tpes of rides in amusement parks in the U.S. in 003, 00, and 005. Use the table to eplain whether each statement is supported b the data. (Lesson 1-1) Total Children s Rides Famil and Adult Rides Roller Coasters Source: National Safet Council a. Injuries on children s rides decreased slightl each ear. b. The number of injuries on roller coasters decreased from 003 to 00. c. Roller coasters are not as safe as children s rides.. Pizza π restaurant made 300 pizzas esterda; % of the pizzas had no toppings, 10% of the pizzas had two toppings, and % had more than two toppings. How man pizzas had at least two toppings? (Previous Course) EXPLORATION 3. a. Eplore g() = b( ) for b < 0. Eplain what happens to the graph of g as b changes. b. Eplore h() = ( _ a ) ( _ a ) - ( _ a ) for a < 0. Eplain what happens to the graph of h as a changes. QY ANSWER ( ); a vertical scale change of magnitude 1 Transformations of Graphs and Data