Homework, Page 135

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

Homework, Page 135

Find the (x, y) pair for the value of the parameter.

1.

23 and 5 for 2x t y t t

23 2 2 5 6,9x y


Homework, Page 135

Find the (x, y) pair for the value of the parameter.

3.

3 4 and 1 for 3x t t y t t

3

3

4 and 1 for 3

3 4 3 =15; 3 1 2 15,2

x t t y t t

x y


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. 2 and 3 1x t y t


Homework, Page 135

5. a. Find the points determined by t = –3, –3, –1, 0, 1, 2, 3.

2 and 3 1x t y t t x y

–3 –6 –10

–2 –4 –7

–1 –2 –4

0 0 –1

1 2 2

2 4 5

3 6 8


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.

2 and 3 1

33 1 1

2 2

x t y t

x xy y


Homework, Page 135

5. c. Graph the relationship in the xy-plane.

2 and 3 1x t y t

x

y


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.

x

y


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

13. 3 6f x x

1

3 6

3 6

3 6

6 3

6 6

3 3

Domain : :

f x x

y x

x y

x y

x xy f x

x x



17. 3f x x

2

2 1 2

3

3

3

3

3 3

: : 0

f x x

y x

x y

x y

y x f x x

Domain x x



21. 3 5f x x

3

3

3

3

3 1 3

5

5

5

5

5 5

Domain : :

f x x

y x

x y

x y

y x f x x

x x


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 ofthe function about y = x .

x

y


Homework, Page 135

Confirm that f and g are inverses by showing that

29.

andf g x x

3 31 and 1f x x g x x

.g f x x

33

3 33 3 33

1 1 1 1

1 1 1 1

f g x x x x

g f x x x x x


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?


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).

1.08 1.08 100 108 eurosy x y y

1.08 1.081.08

xy x x y y


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?

48$44.44

1.08 1.08

xy y y


Homework, Page 135

37. Which basic function can be defined parametrically as follows?

3 6 and for x t y t t

2 for y x y x x


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)

2 5 9x y y

22

22

5 9 2 5 9 9 9 1 2,1

5 9 2 5 9 9 9 1 2,1

x y y y y y y

x y y y y y y


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?

2110 cos 60 and 110 sin 60 16x t y t t


Homework, Page 135

49. a. Graph the function per the instructions. Does the ball clear the fence?

The ball does not clear the

fence.

2110 cos 60 and 110 sin 60 16x t y t t


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.

2110 cos 30 and 110 sin 30 16x t y t t


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.

2110 cos 45 and 110 sin 45 16x t y t t

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1.6

Graphical Transformations


Quick Review

2

2

2

2

3

Write the expression as a binomial squared.

1. 4 4

2. 2 1

3. 4 36 81

Perform the indicated operations and simplify.

4. ( 1) ( 1) 2

5. ( 1) ( 1) 2

x x

x x

x x

x x

x x


Quick Review Solutions

2

2

2

3

2

2

22

2

Write the expression as a binomial squared.

1. 4 4

2. 2 1

3. 4 36 81

Perform the indicated operations and simplify.

4.

( 2)

( 1)

(2 9)

2( 1) ( 1) 2

5. ( 1) ( 1)

x x

x x

x x

x x

x

x

x x

x

x

x

3 22 4 3x x x


What you’ll learn about

Transformations Vertical and Horizontal Translations Reflections Across Axes Vertical and Horizontal Stretches and Shrinks Combining Transformations

… and whyStudying transformations will help you to understand the relationships between graphs that have similarities but are not the same.


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


Translations

Let c be a positive real number. Then the followingtransformations result in translations of the graph of

y=f(x).Horizontal Translations

y=f(x–c) a translation to the right by c unitsy=f(x+c) a translation to the left by c units

Vertical Translationsy=f(x)+c a translation up by c unitsy=f(x)–c a translation down by c units


Example Vertical TranslationsDescribe how the graph of ( ) | | can be transformed

to the graph of | | 4.

f x x

y x


Example Finding Equations for Translations

31

2 2

Each view shows the graph of and a vertical

or horizontal translation . Write an equation for .

y x

y y


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)


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.)


Compositions With Absolute Value

Match the compositions of y = f(x) with the graphs.

1. 2.

3. 4.

a. b. c.

d. e. f.

x

y

y f x y f x

y f x y f x

x

y

x

y

x

y

x

y

x

y

x

y


Stretches and Shrinks

a stretch by a factor of if > 1

a shrink by a factor of if < 1

c cxy f

c cc

a stretch by a factor of if > 1

a shrink by a factor of if < 1

c cy c f x

c c

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


Example Finding Equations for Stretches and Shrinks

31 1

1

2 1

3 1

Let be the curve defined by 3. Find equations

for the following non-rigid transformations of :

(a) is a vertical stretch of by a factor of 4.

(b) is a horizontal shrink of by a f

C y x

C

C C

C C

actor of 1/3.


Example Combining Transformations in Order

2The graph of undergoes the following

transformations, in order. Find the equation

of the graph that results.

a horizontal shift 5 units to the left

a vertical stretch by a factor of 3

a verti

y x

cal translation 4 units up


Homework

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

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

1.7

Modeling with Functions


Quick Review

2

3

Solve the given formula for the given variable.

11. Solve for :

21

2. Solve for : 3

43. Solve for :

34.

b A bh

h V r h

r V r

Area of a Triangle

Volume of a Right Circular Cylinder

Volume of a Sphere

Surface 2 Solve for : 4

5. Solve for :

r A r

P I Prt

Area of a Sphere

Simple Interest


Quick Review Solutions

2

2

Solve the given formula for the given variable.

11. Solve for :

21

2. Solve for : 3

3.

2

3

Solve for

Ab

hb A bh

hV

hV r hr

Area of a Triangle

Volume of a Right Circular Cylinder

Volume of a Sphere 33

2

4:

3

4. Solve for :

3

4

4 4

5. Solve for :

r V r

r A r

P I Prt

Vr

Ar

IP

rt

Surface Area of a Sphere

Simple Interest


What you’ll learn about

Functions from Formulas Functions from Graphs Functions from Verbal Descriptions Functions from Data

… and whyUsing a function to model a variable under observation in terms of another variable often allows one to make predictionsin practical situations, such as predicting the future growth of a business based on data.


Example A Maximum Value Problem

A square of side inches is cut out of each corner of an 8 in. by 15 in. piece

of cardboard and the sides are folded up to form an open-topped box.

(a) Write the volume as a function of .

(b) Find th

x

V x

e domain of as a function of .

(c) Graph as a function of over the domain found in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold.

(d) How b

V x

V x

ig should the cut-out squares be in order to produce the box of

maximum volume?





(a) Write the volume as a function of .

x

V x





(b) Find the domain of as a function of .

x

V x





(c) Graph as a function of over the domain found

x

V x in part (b) and use

the maximum finder on your grapher to determine the maximum volume

such a box can hold. Use a window of [0, 5] by [0, 100].





(d) How big should the cut-out squares be in order to

x

produce the box of

maximum volume?


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?


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:

1. Make a scatter plot of the data points. The points do not need to pass the vertical line test.

2. 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.).

3. Transform a basic function of that type to fit the points as closely as possible.


Models from DataThe 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 City Low High

1 70 86 11 76 92

2 62 80 12 70 85

3 52 72 13 50 70

4 70 94 14 67 89

5 68 86 15 64 88

6 61 86 16 57 79

7 82 106 17 62 90

8 64 90 18 60 86

9 65 87 19 52 68

10 54 76 20 66 84


Models from DataFind a function that relates temperatures and predict the high for

a low of 46.


Functions


Functions (cont’d)

Documents

Homework, Page 135