1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 1 Homework, Page 124 Find the formulas for f + g, f – g, and fg. Give the domain of each. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 2 Homework, Page 124 Find the formulas for f / g, and g / f. Give the domain of each. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 3 Homework, Page 124 9. and are shown in a[0. 5] by [0, 5] viewing window. Sketch the graph of the sum

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 4 Homework, Page 124 Find and 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 5 Homework, Page 124 Find and and find the domain of each. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 6 Homework, Page 124 Find and and find the domain of each. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 7 Homework, Page 124 Find and so that the function can be described as. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 8 Homework, Page 124 Find and so that the function can be described as. 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 9 Homework, Page 124 33. A satellite camera takes a rectangular- shaped picture. The smallest region that can be photographed is a 5-km by 7-km rectangle. As the camera zooms out, the length and width of the rectangle increase at the rate of 2 km/sec. How long does it take for the area A to be at least five times the original size?

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 10 Homework, Page 124 33.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 11 Homework, Page 124 Find two functions defined implicitly by the given relations. 37.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 12 Homework, Page 124 Find two functions implicitly defined by the given relations. 41.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 13 Homework, Page 124 45. True or False. The domain of the quotient function consists of all numbers that belong to both the domain of f and the domain of g. Justify your answer. False. The domain of the quotient function consists of all numbers that belong to both the domain of f and the domain of g, less those that result in.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 14 Homework, Page 124 49. If, then a.b. c.d. e.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 15 Homework, Page 124 53. An identity for a function operation is a function that combines with a given function f to return the same function f. Find the identity functions for the following operations. a. Find a function g such that b. Find a function g such that c. Find a function g such that

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 16 Homework, Page 124 53. a. Find a function g such that b. Find a function g such that

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 17 Homework, Page 124 53. c. Find a function g such that

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.5 Parametric Relations and Inverses

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 19 Quick Review

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 20 Quick Review Solutions

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 21 What you’ll learn about Defining Relations Parametrically Inverse Relations Inverse Functions … and why Some functions and graphs can best be defined parametrically, while some others can be best understood as inverses of functions we already know.

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 22 Parametric Equations A parametric equation is one that defines the two elements of an ordered pair in terms of a third variable, called the parameter. A pair of parametric equations may define either a function or a relation. Remember, a function is a relation that passes the vertical line test.

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 23 Example Defining a Function Parametrically

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 24 Example Defining a Function Parametrically

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 25 Inverse Relation The ordered pair (a,b) is in a relation if, and only if, the pair (b,a) is in the inverse relation.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 26 Horizontal Line Test The inverse of a relation is a function if, and only if, any horizontal line intersects the graph of the original relation at no more than one point. A function that passes this horizontal line test is called a one-to-one function.

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 27 Inverse Function

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 28 Example Finding an Inverse Function Algebraically

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 29 The Inverse Reflection Principle The points (a, b) and (b, a) in the coordinate plane are symmetric with respect to the line y = x. The points (a, b) and (b, a) are reflections of each other across the line y = x.

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 30 The Inverse Composition Rule

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 31 Example Verifying Inverse Functions

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 32 How to Find an Inverse Function Algebraically

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 33 Example Finding Inverse Functions

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 34 Homework Homework Assignment #5 Review Section 1.6 Page 135, Exercises: 1 – 49 (EOO), skip 45 Quiz next time

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1.6 Graphical Transformations

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 36 Quick Review

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 37 Quick Review Solutions

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 38 What you’ll learn about Transformations Vertical and Horizontal Translations Reflections Across Axes Vertical and Horizontal Stretches and Shrinks Combining Transformations … and why Studying transformations will help you to understand the relationships between graphs that have similarities but are not the same.

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 39 Transformations Rigid transformation – an action that changes a graph in a predictable manner. The shape and the size of the graph remain unchanged, but its position changes horizontally, vertically or diagonally. Non-rigid transformation – generally a distortion of the shape of a graph, including horizontal or vertical stretches and shrinks

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 40 Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y=f(x). Horizontal Translations y=f(x-c)a translation to the right by c units y=f(x+c)a translation to the left by c units Vertical Translations y=f(x)+ca translation up by c units y=f(x)-ca translation down by c units

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 41 Example Vertical Translations

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 42 Example Finding Equations for Translations

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 43 Reflections The following transformations result in reflections of the graph of y = f(x): Across the x-axis y = – f(x) Across the y-axis y = f(– x)

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 44 Graphing Absolute Value Compositions Given the graph of y = f(x), the graph y = |f(x)| can be obtained by reflecting the portion of the graph below the x-axis across the x-axis, leaving the portion above the x-axis unchanged; the graph of y = f(|x|) can be obtained by replacing the portion of the graph to the left of the y-axis by a reflection of the portion to the right of the y-axis across the y-axis, leaving the portion to the right of the y-axis unchanged. (The result will show even symmetry.)

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 45 Compositions With Absolute Value Match the compositions of y = f(x) with the graphs. 1.2. 3.4. a.b.c. d.e.f.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 46 Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph y = f(x). Horizontal Stretches or Shrinks Vertical Stretches and Shrinks

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 47 Example Finding Equations for Stretches and Shrinks

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 1- 48 Example Combining Transformations in Order