2003 Joe Vasta

Modeling with Functions – Homework

1. A rectangle has perimeter 100 feet. Find a function that models its area A in

terms of its length l .

2. A rectangle has area 20 square feet. Find a function that models its perimeter

P in terms of its width w.

3. A rectangle has width 4 feet. Find a function that models its area A in terms

of its perimeter P.

4. A rectangle has length 5 feet. Find a function that models its perimeter P in

terms of its area A.

5. The length of a rectangle is 5 feet longer than the width. Find a function that

models its area A in terms of its perimeter P.

6. The length of a rectangle is 5 times the width. Find a function that models its

perimeter P in terms of its area A.

7. The short leg of a right triangle is 4 feet. Find a function that models its area

A in terms of its hypotenuse z.

8. The long leg of a right triangle is 5 feet. Find a function that models its

perimeter P in terms of its short leg x.

9. The hypotenuse of a right triangle is 10 feet. Find a function that models its

area A in terms of its short leg x.

10. The hypotenuse of a right triangle is 6 feet. Find a function that models its

perimeter P in terms of its long leg y.

11. The long leg of a right triangle is 5 times the short leg. Find a function that

models its area A in terms of its hypotenuse z.

12. The hypotenuse of a right triangle is 6 inches longer than the short leg. Find a

function that models its perimeter P in terms of its short leg x.

13. Find a function that models the area A of a circle in terms of the

circumference C.

14. Find a function that models the circumference C of a circle in terms of the

area A.

15. Find a function that models the area A of a square in terms of the diagonal d.

16. Find a function that models the diagonal d of a square in terms of the

perimeter P.

2003 Joe Vasta

Modeling with Functions – Homework Answers

1. 250)( lllA

2. ww

wP 240

)(

3. 162)( PPA

4. 5

210)(

AAP

5. 16

100)(

2

pPA

6. 5

12)(

AAP

7. 162)( 2 zzA

8. 255)( 2 xxxP

9. 2

100)(

2xxxA

10. 2366)( yyyP

11. 52

5)(

2zzA

12. 93262)( xxxP

13. 4

)(2C

CA

14. AAC 2)(

15. 2

)(2d

dA

16. 4

2)(

PPd

2012 Joe Vasta

Transforming The Square Root Function – Homework

Graph the function. Find the x- and y-intercepts if they exist.

1. ( ) 3f x x

2. ( ) 3f x x

3. ( ) 2f x x

4. ( ) 2f x x

5. 1

( ) 3 32

f x x

6. ( ) 1 2f x x

2012 Joe Vasta

Transforming The Square Root Function – Homework Answers

1. x-int: (3, 0); y-int: none 2. x-int: (9, 0); y-int: (0, –3)

x

y

x

y

3. x-int: (0, 0); y-int: (0, 0) 4. x-int: (0, 0); y-int: (0, 0)

x

y

x

y

5. x-int: (33, 0); y-int: (0, 3 / 2 3) 6. x-int: (1/2, 0); y-int: (0, –1)

x

y

x

y

2006 Joe Vasta

Absolute Value Functions – Homework

a. Rewrite each absolute value function as a piecewise defined function.

b. Graph each function.

1. ( )f x x

2. ( )x

f xx

3. ( )f x x x

4. 2

( )x x

f xx

5. ( ) 3f x x

6. 2 1

( )1

xf x

x

7. ( ) 3 2f x x x

8. 2 4

( ) 3 44

xf x x x

x

2006 Joe Vasta

Absolute Value Functions – Homework Answers

1. if 0

( ) if 0

x xf x

x x

x

y

2. 1 if 0

( )1 if 0

xf x

x

x

y

3. 2

2

if 0( )

if 0

x xf x

x x

x

y

4. 3 if 0

( )1 if 0

xf x

x

x

y

5. 3 if 3

( )3 if 3

x xf x

x x

x

y

6. 2 if 1

( )2 if 1

xf x

x

x

y

7. 5 if 2

( )1 2 if 2

xf x

x x

x

y

8. 2 5 if 4

( )1 if 4

x xf x

x

x

y

Graphs with Holes – Homework

Graph each function neatly. Label all intercepts and holes.

1. f(x) = 3

62

x

xx 2. g(x) =

4

24102

x

xx

3. h(x) = x

xx 22 4. j(x) =

12

472 2

x

xx

5. k(x) = 23

10116 2

x

xx 6. n(x) =

4

16 2

x

x

7. p(x) = 3

3

x

x 8. q(x) =

34

332

23

xx

xxx

9. s(x) = xx

xxx

23

61362

23

10. f(x) =

1

4423

x

xxx

11. g(x) = x

xx 43 12. h(x) =

2

2 23

x

xx

13. j(x) = 1

22 23

x

xx 14. k(x) =

xx

x

2

22

15. m(x) = 23 3

3

xx

x

2002 Joe Vasta

x

y

x

y

y

x

Graphs with Holes – Homework Answers 1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

13. 14. 15.

x

y

x

y

x

y

x

y

x

y y

x

y

x

x

y y

x

x

y

x

y y

x

2012 Joe Vasta

Inverse Relations – Homework

Classify each relation as (1) not a function, (2) function, but not 1-1, or (3) 1-1 function.

Draw the inverse relation.

1. 2. 3.

x

y

x

y

x

y

4. 5. 6.

x

y

x

y

x

y

7. 8. 9.

x

y

x

y

x

y

2012 Joe Vasta

Inverse Relations – Homework Answers

1. 2. 3.

a. 1-1 function a. not a function a. function, but not 1-1

b. b. b.

x

y

x

y

x

y

4. 5. 6.

a. function, but not 1-1 a. 1-1 function a. function, but not 1-1

b. b. b.

x

y

x

y

x

y

7. 8. 9.

a. not a function a. not a function a. 1-1 function

b. b. b.

x

y

x

y

x

y

2000 Joe Vasta

Type 1 Exponential Equations – Homework

Solve.

1. 2 64x 2. 3 81x 3. 51222

x

4. 13 27x 5. 2 15 125x 6. 25625 x

7. 4972

xx 8. 12

xxb 9. 322

1

x

10. 000,1010

1

x 11. 9 3x 12. 64 8x

13. 9 27x 14. 64 16x 15. 749

1

x

16. 125

15 x 17.

4

9

8

27

x

18. 0.01 1000x

19. 1

12

9

7293

x

x 20. xxx 1422

21. 122 2

273 xx

2000 Joe Vasta

Type 1 Exponential Equations – Homework Answers

1. 6 2. 4 3. 3

4. 4 5. 1 6. 8/5

7. –2, 1 8. 0, –1 9. –5

10. –4 11. 1/2 12. 1/2

13. 3/2 14. 2/3 15. –1/2

16. –3 17. 2/3 18. –3/2

19. 5/4 20. –1, 2 21. –1, 5/3

2012 Joe Vasta

Parametric Equations – Homework

Eliminate the parameter to find a rectangular equation of the curve.

Sketch the curve and indicate the orientation.

1. x = –t + 1, y = –2t + 6, 1 ≤ t ≤ 3

2. x = 2t – 12, y = 4t – 20, 5 ≤ t ≤ 6

3. x = t – 1, y = 3 – t, 2 ≤ t ≤ 6

4. x = –1

2t + 4, y = –

1

2t + 3, 4 ≤ t ≤ 10

5. x = 2t – 4, y = t – 2, t ≥ 3

6. x = 7 – 4t, y = 8t – 12, t ≥ 1

7. x = t + 4, y = t2 + 1, –1 ≤ t ≤ 2

8. x = 1 – t, y = t2, –1 ≤ t ≤ 2

9. x = t – 3, y = (t – 2)2, t ≥ 0

10. x = 2 – t, y = –t2, t ≥ 0

11. x = t2, y = t + 1

12. x = t2 + 1, y = –t

2012 Joe Vasta

Parametric Equations – Homework Answers

1. y = 2x + 4, –2 ≤ x ≤ 0 2. y = 2x + 4, –2 ≤ x ≤ 0 3. y = –x + 2, 1 ≤ x ≤ 5

x

y

x

y

x

y

4. y = x – 1, –1 ≤ x ≤ 2 5. y = x/2, x ≥ 2 6. y = –2x + 2, x ≤ 3

x

y

x

y

x

y

7. y = (x – 4)2 + 1, 3 ≤ x ≤ 6 8. y = (1 – x)

2, –1 ≤ x ≤ 2 9. y = (x + 1)

2, x ≥ –3

x

y

x

y

x

y

10. y = –(2 – x)2, x ≤ 2 11. (y – 1)

2 = x 12. y

2 = x – 1

x

y

x

y

x

y

2003 Joe Vasta

Partial Fractions – Homework

Find the partial fraction decomposition of the rational expression.

1. 2

8 2

2 8

x

x x

2.

xx

x

5

3072

3. 2

2 1

4 3

x

x x

4.

26

9242

xx

x

5. 2

2 1

2 1

x

x x

6.

44

1072

xx

x

7. 2

12

x

x 8.

144

5162

xx

x

9. 2

3 2

4 14 6

2 3

x x

x x x

10.

2

3

6 8 8x x

x x

11. 2

3 2

5 11 7x x

x x

12.

2

3 2

7 13 6

2 4 8

x x

x x x

13. 2

3 2

3 4 5

1

x x

x x x

14.

2

3 2

3 5 7

4 4

x x

x x x

15. 2

2

2 4 58

12

x x

x x

16.

3 2

2

9 19 6

8 16

x x x

x x

2003 Joe Vasta

Partial Fractions – Homework Answers

1. 4

5

2

3

xx 2.

6 1

5x x

3. 1 5

2( 1) 2( 3)x x

4.

23

3

12

6

xx

5. 2)1(

3

1

2

xx 6.

2)2(

4

2

7

xx

7. 2

12

xx

8.

2)12(

3

12

8

xx

9. 2 3 1

1 3x x x

10.

8 3 5

1 1x x x

11. 2

1 4 7

1x x x

12.

2

3 4 1

2 2 ( 2)x x x

13. 2

2 3

1 1

x

x x

14.

2

1 2 3

1 4

x

x x

15. 4 6

23 4x x

16. 2

5 21

4 ( 4)x

x x