1. Homework, Page 135
Find the (x, y) pair for the value of the parameter. 1.

2. Homework, Page 135
Find the (x, y) pair for the value of the parameter. 3.

3. Homework, Page 135
a. Find the points determined by t = –3, –2, –1, 0, 1, 2, 3.
b. Find a direct algebraic relationship between x and y and determine whether the parametric equations determine y as a function of x.
c. Graph the relationship in the xy-plane. 5.

4. 5. a. Find the points determined by t = –3, –3, –1, 0, 1, 2, 3.
Homework, Page 135
5. a. Find the points determined by t = –3, –3, –1, 0, 1, 2, 3. t x y –3 –6
–10 –2 –4 –7 –1 1 2 4 5 3 6 8

5. Homework, Page 135
5. b. Find a direct algebraic relationship between x and y and determine whether the parametric equations determine y as a function of x.
The relation yields y as a function of x, as the relation passes the vertical line test.

6. Homework, Page 135
5. c. Graph the relationship in the xy-plane.

7. Homework, Page 135
The graph of a relation is shown.
a. Is the relation a function?
b. Does the relation have an inverse that is a function? 9.
a. The relation is not a
function.
b. The relation has an
inverse that is a function.

8. Homework, Page 135
Find a formula for f –1(x). Give the domain of f –1, including any restrictions ‘inherited’ from f.
13.

9. Homework, Page 135
Find a formula for f –1(x). Give the domain of f –1, including any restrictions ‘inherited’ from f.
17.

10. Homework, Page 135
Find a formula for f –1(x). Give the domain of f –1, including any restrictions ‘inherited’ from f.
21.

11. Homework, Page 135
Determine if the function is one-to-one. If it is, sketch the graph of the inverse.
25.
The function is one-
to-one, and the
inverse function
graphs as shown,
as a reflection of
the function about
y = x .

12. Homework, Page 135 Confirm that f and g are inverses by showing that
29.

13. Homework, Page 135
33. In May 2002, the exchange rate for converting U.S. dollars (x) into euros (y) was y = 1.08x.
a. How many euros could you get for $100 U.S?
b. What is the inverse function and what conversion does it represent?
c. In the spring of 2002, a tourist had an elegant lunch in Provence, France ordering from a “fixed price” 48-euro menu. How much was that in U.S. dollars?

14. Homework, Page 135 33. a. How many euros could you get for $100 U.S?
b. What is the inverse function and what conversion does it represent?
The inverse function represents the exchange rate for converting euros (x) into U.S. dollars (y).

15. Homework, Page 135
33. c. In the spring of 2002, a tourist had an elegant lunch in Provence, France ordering from a “fixed price” 48-euro menu. How much was that in U.S. dollars?

16. Homework, Page 135
37. Which basic function can be defined parametrically as follows?

17. Homework, Page 135
41. Which ordered pair is in the inverse of the relation given by ?
A. (2, 1) B. (–2, 1)
C. (–1, 2) D. (2, –1)
E. (1, –2)

18. Homework, Page 135
49. A baseball that leaves the bat at an angle of 60º above the horizontal with a speed of 110 ft/sec may be modeled by the following parametric equations:
a. Graph the function per the instructions. Does the ball clear the fence?
b. Does the ball clear the fence at 30º?
c. What angle is optimum for hitting the ball? Does it clear the fence when hit at that angle?

19. Homework, Page 135
49. a. Graph the function per the instructions. Does the ball clear the fence?
The ball does not clear the
fence.

20. Homework, Page 135
49. b. Does the ball clear the fence at 30º?
The ball does not clear the
fence at 30º; it barely
reaches it.

21. Homework, Page 135
49. c. What angle is optimum for hitting the ball? Does it clear the fence when hit at that angle?
The optimum angle for
hitting the ball is 45º, and
the ball does clear the fence
when hit at this angle.

22. Graphical Transformations
1.6
Graphical Transformations

23. Quick Review

24. Quick Review Solutions

25. 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.

26. 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

27. 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)+c a translation up by c units
y=f(x)–c a translation down by c units

28. Example Vertical Translations

29. Example Finding Equations for Translations

30. 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)

31. 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.)

32. Compositions With Absolute Value
Match the compositions of y = f(x) with the graphs.
a. b. c.
d. e. f.

33. 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

34. Example Finding Equations for Stretches and Shrinks

35. Example Combining Transformations in Order

36. Homework
Review Section 1.6
Page 147, Exercises: 1 – 65 (EOO)

37. Modeling with Functions
1.7
Modeling with Functions

38. Quick Review

39. Quick Review Solutions

40. What you’ll learn about
Functions from Formulas
Functions from Graphs
Functions from Verbal Descriptions
Functions from Data
… and why
Using a function to model a variable under observation
in terms of another variable often allows one to make predictions
in practical situations, such as predicting the future growth of a
business based on data.

41. Example A Maximum Value Problem

42. Example A Maximum Value Problem

43. Example A Maximum Value Problem

44. Example A Maximum Value Problem

45. Example A Maximum Value Problem

46. Example Finding the Model and Solving
Grain is leaking through a hole in a storage bin at a constant rate of 5 cubic inches per minute. The grain forms a cone-shaped pile on the ground below. As it grows, the height of the cone always remains equal to its radius. If the cone is one foot tall now, how tall will it be in one hour?

47. Constructing a Function from Data
Given a set of data points of the form (x,y), to construct
a formula that approximates y as a function of x:
Make a scatter plot of the data points. The points do not need to pass the vertical line test.
Determine from the shape of the plot whether the points seem to follow the graph of a familiar type of function (line, parabola, cubic, sine curve, etc.).
Transform a basic function of that type to fit the points as closely as possible.

48. Models from Data
The table shows low and high temperatures in 20 cities. Find a
function that relates temperatures and predict the high for a low
of 46.
City
Low
High 1 70 86 11 76 92 2 62 80 12 85 3 52 72 13 50 4 94 14 67 89 5 68 15 64 88 6 61 16 57 79 7 82
106 17 90 8 18 60 9 65 87 19 10 54 20 66 84

49. Models from Data
Find a function that relates temperatures and predict the high for a low of 46.

50. Functions

51. Functions (cont’d)