Shifting, Reflecting, and Stretching Graphs

Transcription

1 Shifting, Reflecting, and Stretching s Shifting s 1 ( ) ( ) This is f ( ) This is f ( ) This is f ( ) What happens to the graph? f ( ) is f () shifted units to the right. f ( ) is f () shifted units to the left. Turn off and. 5 This is f ( ) This is f ( ) What happens to the graph? f ( ) is f () shifted units up. f ( ) is f () shifted units down. 1

2 Vertical and Horizontal Shifts Let c be a positive real number. The following changes in the function f () will produce the stated shifts in the graph of f () 1. h( ) f ( c) Horizontal shift c units to the right. h( ) f ( c) Horizontal shift c units to the left. h( ) c Vertical shift c units downward. h( ) c Vertical shift c units upward Eample: Given f() = +, describe the shifts of the graph of f generated b the following functions. a) g() = ( + 1) Horizontal shift 1 unit to the left. b) h() = ( - ) + Horizontal shift units to the right.

3 Eample: Let f() =. Write the equation for the function resulting from a vertical shift of units downward and a horizontal shift of units to the right of the graph of f() =. Answer: f() = - -. Reflecting s 1 ( ) ( ) ( ) Note that this is f (). Note that this is f ( ). What happens to the graph? f () is reflected in the -ais f ( ) is reflected in the -ais The following changes in f () will produce the stated reflections of the graph of f (). 1. h( ) : reflection in the -ais. h( ) f ( ) : reflection in the -ais

4 Eample: Let f ( ) Describe the graph of g( ) in terms of f. The graph of g is a reflection of the graph of f in the -ais. Definition: A rigid transformation is a transformation in which the basic shape of the graph is unchanged. Rigid transformations change onl the position of the graph in the -plane. Three tpes of rigid transformations: 1. Horizontal shifts. Vertical shifts. Reflections Nonrigid Transformations 1 f ) 8 ( Note that this is 8 f ( ) 1 1 Note that this is f ( )..

5 What happens to the graph? 8 appears narrower than 1 appears flatter than Definition: A non-rigid transformation is a transformation that actuall distorts the shape of a graph instead of just shifting or reflecting it. From our eample: 8 is called a vertical stretch 1 is called a vertical shrink 1 f ( ) () Note that this is ( ) 1 1 ( Note that this is ( ) ) f. f. What happens to the graph? ( ) appears steeper than 1 ( ) appears wider than 5

6 Consider the graph of f(): f() Now look at f(): f()

7 This is a vertical stretch b a factor of. Notice that the horizontal aspect of the graph has not changed. Now look at f(): f() -1 -½ 0 ½ 0 1 This is a horizontal shrink b a factor of. Notice that the vertical aspect of the graph has not changed. In general, for f () and the real number c, A vertical stretch is written g( ) cf ( ), where c 1 A vertical shrink is written g( ) cf ( ), where 0 < c < 1 A horizontal shrink is written h( ) f ( c), where c 1 A horizontal stretch is written h( ) f ( c),where 0<c< 1 7

8 Eample: Compare the graph of each function with the graph of f() = +. (a) g() = f() g() = f() = + () = + This is a horizontal shrink of the graph of f(). (b) h() = f( 1 ) h() = f( 1 ) = + ( 1 ) = This is a horizontal stretch of the graph of f(). Eample: Compare each graph with the graph of f() = (a) g() = - A reflection of f in the -ais, since g() = -f(). (b) h() = A reflection of f in the -ais, since h() = f(-). 8

9 Eample: Compare each graph with the graph of f() = (c) k() = A shift of f units left, followed b a reflection in the -ais, since k() = -f(+) (d) p() = A horizontal shrink, since p( ) f (). Consider j() = 1 + compared to the graph of f ( ) 1 1 ( 1) (shift the graph 1 unit left) (shift the graph unit up) (vertical stretch of ) 5 1 ( 1) (shift the graph 1 unit left) 6 5 (vertical stretch of ) (shift the graph unit up) 7 5 9

10 Turn off all graphs ecept and 7. The graphs are not the same. To see which is correct, 1 8 Which graph matches the correct answer? Answer: 7 Conclusion: Alwas follow the order of operations when considering transformations. Summar In general, for f () and the real number c, h( ) f ( c) Horizontal shift c units to the right h( ) f ( c) Horizontal shift c units to the left h( ) c Vertical shift c units downward h( ) c Vertical shift c units upward h( ) Reflection in the -ais h( ) f ( ) Reflection in the -ais h( ) cf ( ) Vertical stretch, where c 1 h( ) cf ( ) Vertical shrink, where 0 < c < 1 h( ) f ( c) Horizontal shrink, where c 1 h( ) f ( c) Horizontal stretch, where 0 <c < 1 10