1. 1.4.1 MATHPOWER TM 12, WESTERN EDITION Chapter 1 Transformations 1.4

2. f(x) = | x | f(x) = 2 | x | f(x) = 3 | x | 0 1 2 3 1.4.2 Vertical Stretches of Functions

3. f(x) = | x | 1.4.3 Vertical Stretches of Functions [cont’d] A stretch can be an A stretch can be

4. Given the graph of y = f(x), there is a 1.4.4 Graphing y = af(x) y = | x |

5. Given the graph of y = f(x), there is a 1.4.5 Graphing y = af(x) y = | x |

6. In general, for any function y = f(x), the graph of a function y = af(x) is obtained by multiplying the y-value of each point on the graph of y = f(x) by a. That is, the point (x, y) on the graph of y = f(x) is transformed into the point (x, ay) on the graph of y = af(x). If a < 0, the graph is in the x-axis. 1.4.6 Vertical Stretching and Reflecting of y = f(x)

7. y = f(x) For y = af(x), there is a vertical stretch. 1.4.7 Vertical Stretching and Reflecting of y = f(x)

8. f(x) = x 2 f(x) = (2x) 2 0 1 4 9 f(x) = (0.5x) 2 Each point on the graph of y = (2x) 2 is half as far from the y-axis as the related point on the graph of y = x 2. The graph of y = f(2x) is a of the graph of y = f(x) by a factor of Each point on the graph of y = (0.5x) 2 is as the related point on the graph of y = x 2. The graph of y = f(0.5x) is a of the graph of y = f(x) by a factor of 1.4.8 Horizontal Stretching of y = f(x)

9. y = x 2 (-1, 1) (1, 1) For y = f(kx), there is a Horizontal Stretching of y = f(kx) when k > 1 1.4.9

10. (-1, 1) (1, 1) y = x 2 For y = f(kx), there is a 1.4.10 Horizontal Stretching of y = f(kx) when 0 < k < 1

11. In general, for any function y = f(x), the graph of the function y = f(kx) is obtained by at each point on the graph of y = f(x) by k. That is, the point (x, y) on the graph of the function y = f(x) is transformed into the point on the graph of y = f(kx). If k < 0, there is also a in the. Comparing y = f(x) With y = f(kx) 1.4.11

12. y = f(x) Graph y = f(2x). 1.4.12 Graphing y = f(kx) and its Reflection

13. Describe what happens to the graph of a function y = f(x). a) y = f(3x) b) 3y = f(x) c) y = f( x)d) -2y = f(x) e) 2y = f(2x) f) y = f(-3x) 1.4.13 Describing the Horizontal or Vertical Stretch of a Function

14. The graph of the function y = f(x) is transformed as described. Write the new equation in the form y = af(kx). a) Horizontal stretch factor of one-third, and a vertical stretch factor of two b) Horizontal stretch factor of two, a vertical stretch by a factor of one-third, and a reflection in the x-axis c) Horizontal stretch factor of one-fourth, a vertical stretch factor of three, and a reflection in the y-axis d) Horizontal stretch factor of three, vertical stretch factor of one-half, and a reflection in both axes 1.4.14 Stating the Equation of y = af(kx)

15. Given the graph of sketch the graphs with the following transformations. a) Stretch horizontally by a factor of 2. y16x 4  2  = 1.4.15,

16. State the zeros of this polynomial, and a possible equation of P(x). Graphing a Polynomial and its Transformations 1.4.16

17. Suggested Questions: Pages 38-40 1-26, 27-41, 45-51 1.4.17