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Stretches of Functions

Chapter 1 Transformations. 1.4. Stretches of Functions. 1.4. 1. MATHPOWER TM 12, WESTERN EDITION. Vertical Stretches of Functions. f ( x ) = 2 | x |. f ( x ) = | x |. f ( x ) = 3 | x |. 0. 1. 2. 3. 1.4. 2. Vertical Stretches of Functions [cont’d]. f ( x ) = | x |.

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Stretches of Functions

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  1. Chapter 1 Transformations 1.4 Stretches of Functions 1.4.1 MATHPOWERTM 12, WESTERN EDITION

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

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

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

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

  6. Vertical Stretching and Reflecting of y = f(x) 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

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

  8. Horizontal Stretching of y = f(x) f(x) = (2x)2 f(x) = x2 f(x) = (0.5x)2 0 1 4 9 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 = x2. 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 = x2. The graph of y = f(0.5x) is a of the graph of y = f(x) by a factor of 1.4.8

  9. Horizontal Stretching of y = f(kx) when k > 1 y = x2 (-1, 1) (1, 1) For y = f(kx), there is a 1.4.9

  10. Horizontal Stretching of y = f(kx) when 0 < k < 1 y = x2 (-1, 1) (1, 1) For y = f(kx), there is a 1.4.10

  11. Comparing y = f(x) With y = f(kx) 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 . 1.4.11

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

  13. Describing the Horizontal or Vertical Stretch of a Function 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

  14. Stating the Equation of y = af(kx) 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

  15. 2 - ( - ) , Given the graph of 4 y = 16 x sketch the graphs with the following transformations. a) Stretch horizontally by a factor of 2. 1.4.15

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

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

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