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Chapter 6

Chapter 6. 6-4 Transforming Functions. Objectives. Transform functions. Recognize transformations of functions. Transforming functions.

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Chapter 6

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  1. Chapter 6 6-4 Transforming Functions

  2. Objectives Transform functions. Recognize transformations of functions.

  3. Transforming functions • In previous lessons, you learned how to transform several types of functions. You can transform piecewise functions by applying transformations to each piece independently. Recall the rules for transforming functions given in the table.

  4. Transforming functions

  5. Example 1: Transforming Piecewise Functions • Given f(x) = – 1/2x if x < 0 • write the rule g(x),a vertical stretch by a factor of 3. ½ x2if x ≥0

  6. 3(– x) if x < 0 3(x2) if x ≥ 0 – x if x <0 1 1 3 3 x2 if x ≥0 2 2 2 2 Solution • Each piece of f(x) must be vertically stretched by a factor of 3. Replace every y in the function by 3y, and simplify. g(x) = 3f(x) =

  7. x2 if x ≤ 0 Given f(x) = write the rule for g(x), a horizontal stretch of f(x) by a factor of 2. x – 3 if x > 0 Check it out!!

  8. Transforming functions • When functions are transformed, the intercepts may or may not change. By identifying the transformations, you can determine the intercepts, which can help you graph a transformed function

  9. Transforming functions

  10. Transforming functions

  11. Example 2A: Identifying Intercepts • Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts. f(x) =–2x – 4 ; g(x) =

  12. Solution • Find the intercepts of the original function • y-intercept x-intercept –2 = x The y-intercept is –4, and the x-intercept is -2. Note that g(x) is a horizontal stretch of f(x) by a factor of 2. So the y-intercept of g(x) is also –4. The x-intercept is 2(–2), or –4. f(0) = –2(0) – 4 = – 4 0 = –2x – 4

  13. Example 2B: Identify Intercepts • f(x) = x2–1; g(x) = f(–x)

  14. 2 f(x) = x + 4 and g(x) = –f(x) 3 Check it out!!! • Identify the x- and y-intercepts of f(x). Without graphing g(x), identify its x- and y-intercepts. The y-intercept is 4, and the x-intercept is –6. Note that g(x) is a reflection of f(x) across the x-axis. So the x-intercept of g(x) is also –6. The y-intercept is –1(4), or –4.

  15. Example 3: Combining Transformations • Given f(x) = 1/3(x– 2)2 and g(x) = 2f(x) – 3 and graph g(x).

  16. Solution • Step 1 Graph f(x). The graph of f(x)hasy-intercept (0, 4/3) and x-intercept (2, 0).

  17. Solution • Step 2 Analyze each transformation one at a time. • The first transformation is a vertical stretch by a factor of 2. After the vertical stretch, the x-intercept will remain 2, but the y-intercept will be . • The second transformation is a vertical translation of 3 units down. Use a table to shift each identified point down 3 units.

  18. 8 1 3 3 Solution • Intercept Points ( 2, 0) • Shifted (2, –3) Step 3 Graph the final result.

  19. Check it out!!! • Given f(x) = 2x – 4 and g(x)= – 1/2f(x), graph g(x).

  20. Application • A movie theater charges $5 for children under 12 and $7.50 for anyone 12 and over. The theater decides to increase its prices by 20%. It charges an additional $0.50 fee for online ticket purchases. Write an equation for the online ticket prices.

  21. Student guided practice • Do problems 1-7 in your book page 436

  22. Homework • Do problems 8-17 page 436 and 437

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