Chapter 4 Focus Exercise Answers Sec. 4.1 Introduction to Graphing 1.

concave up

Both increasing and decreasing.

2.

concave down

Both increasing and decreasing.

3.

concave up

Increasing only.

4.

concave down

Increasing only.

5.

point of

inflection

concave up

concave down

Increasing only.

6.

point of

inflection

concavedown

concaveup

decreasing only.

7. a) Shifts up 6; 8. a) Shifts down 5; b) is narrower than the standard graph b) is wider than the standard graph 9. a) Shifts right 9; 10. a) Shifts left 4; b) is standard and reflected b) is narrower than the standard graph 11. a) Shifts left 2 and down 1; 12. a) Shifts right 7 and up 3; b) is standard b) is standard and reflected

13. f(x) = 2x2 – 4 y

x2-2-6 6

2

6

6-

2-

4-4

4-

8-

-8 8

4

8

(0, -4)

(1, -2)

(2, 4)

(-1, -2)

(-2, 4)

14. f(x) = -x2 + 6 y

x2-2-6 6

2

6

6-

2-

4-4

4-

8-

-8 8

4

8

(0, 6)

(1, 5)

(2, 2)

(3, -3)

(-1, 5)

(-2, 2)

(-3, -3)

15. f(x) = 2(x + 2)2

y

x2-2-6 6

2

6

6-

2-

4-4

4-

8-

-8 8

4

8

(-2, 0)

(-1, 2)

(0, 8)

(-3, 2)

(-4, 8)

16. f(x) = -(x – 3)2 y

x2-2-6 6

2

6

6-

2-

4-4

4-

8-

-8 8

4

8

(3, 0)

(4, -1)

(5, -4)

(6, -9)

(2, -1)

(1, -4)

(0, -9)

Section 4.2 Graphing the Sine and Cosine Functions 1. f(x) = 4sin(6x)

4

-4

!

12

5!

12

!

6

!

4

!

3

!

2

7!

12

2!

3

2. f(x) = 3cos(2x)

3

-3

!

4

5!

4

!

2

3!

4

3!

2

7!

4

2! !

3. f(x) = -3cos(4x)

3

-3

!

2

!

8

5!

8

!

4

3!

8

3!

4

7!

8

!

4. f(x) = -sin(2x)

1

-1

!

4

5!

4

!

2

3!

4

3!

2

7!

4

2! !

5. f(x) = sin( )1

3x

1

-1

3!

2

3! 9!

2

6! 9! 12!15!

2

21!

2

6. f(x) = cos( )1

2x

1

-1

8!7!6!5!3!2!! 4!

7. f(x) = -2cos( )4

3x

2

-2

3!

2

3!

8

3!

4

9!

8

9!

4

3!15!

8

21!

8

8. f(x) = -3sin( )25x

3

-3

5!

4

5!

2

10!5!15!

4

25!

4

15!

2

35!

4

9. f(x) = 4sin( )52x

4

-4

!

5

2!

5

3!

5

4!

5

6!

5

7!

5

8!

5

!

10. f(x) = 3cos( )83x

1

-1

3!

16

3!

8

9!

16

3!

4

15!

16

9!

8

21!

16

3!

2

11. f(x) = -2cos( )38x

2

-2

4!

3

8!

3

4! 20!

3

28!

3

16!

3

8! 32!

3

12. f(x) = -4sin( )56x

4

-4

3!

5

6!

5

12!

5

21!

5

18!

5

24!

5

9!

5

3!

Sec. 4.3 Find the Function (Note: The answers use f(x) instead of y.) 1. f(x) = 4cos( )1

2x 2. f(x) = sin(2x) 3. f(x) = 5sin(3x)

4. f(x) = 2cos( )83x 5. f(x) = -3sin( )16

5 x 6. f(x) = -53 cos(12x)

7. f(x) = 2cos( )43x 8. f(x) = 5sin( )3

4x 9. f(x) = -52 cos(4x)

10. f(x) = -3sin(6x) 11. f(x) = sin( )8

3x 12. f(x) = cos( )45x

Section 4.4 Graphing with Vertical and Horizontal Shifts 1. f(x) = 2cos(2x) + 2

2

4

!

4

!

2

3!

4

!

2. f(x) = -cos( )32x – 4

!

3

2!

3

4!

3

!

-1

-2

-3

-4

-5

3. f(x) = -2 + 3sin( )13x

3!

2

9!

2

3!

-5

-2

1

6!

4. f(x) = 1 – 2sin( )2

5x

-1

1

3

5!

4

5!

2

5!15!

4

5. f(x) = 3sin( )4x + π2

!

8

!

4

3!

8

3

-!

8

-3

6. f(x) = 4sin( )6x – π2

!

4

!

3

5!

12

4

-4

!

6

!

12

7. f(x) = 4cos 54x –

π4

!

5

3!

5

7!

5

!

-4

4

9!

5

8. f(x) = 4cos 54x –

π4

-!

3

!

3

5!

3

!

-3

3

7!

3

9. f(x) = -sin 23x +

π6

2!5!

4

11!

4

-1

1

-!

4

!

2

10. f(x) = -3sin 43x –

π6

7!

8

-3

3

!

2

!

8

5!

4

13!

8

11. f(x) = -2cos 54x –

π8

13!

10

!

10

9!

10

!

2

2

-2

17!

10

12. f(x) = -2cos 38x +

π16

-!

6

-2

2

31!

6

7!

6

5!

2

23!

6

Sec. 4.5 Find the Function 1. Period = π 2. B = 2 3. f(x) = 3cos( )2x +

π3

4. f(x) = -3cos 2x –

2π3 5. f(x) = -3sin( )2x –

π6 6. f(x) = 3sin( )2x –

7π6

7. Period = 3π 8. B = 23 9. f(x) = -2sin 2

3x – π9

10. f(x) = 2sin 23x +

8π9 11. f(x) = 2cos 2

3x + 7π18 12. f(x) = -2cos 2

3x + 25π18

Section 4.6 Graphing Secant and Cosecant 1. f(x) = csc(2x) 2. f(x) = - sec(2x) 3. f(x) = 2sec(4x)

1

-1

!

4

!

2

3!

4

!

!

2

1

-1

!

4

3!

4

-!

4

!

4

2

-2

!

8

3!

8

-!

8

4. f(x) = -2csc(4x) 5. f(x) = -2csc(3x) 6. f(x) = 2sec(3x)

!

8

2

-2

!

4

3!

8

!

2

!

6

!

2

2

-2

!

3

2!

3

!

3

2

-2

!

6

!

2

-!

6

7. f(x) = -sec(6x)

8. f(x) = csc(6x) 9. f(x) = 3csc( )25x

!

6

1

-1

!

12

!

4

-!

12

1

-1

!

12

!

6

!

4

!

3

3

-3

5!

4

5!

2

5!15!

4

10. f(x) = -3sec( )2

5x 11. f(x) = - 32 sec( )1

3x 12. f(x) = 32 csc( )1

3x

3

-3

5!

4

-5!

4

5!

2

15!

4

3!3!

2

9!

2

2

-1

1

-2

-3!

2

2

-2

3!

2

6!

1

-19!

2

3!

Section 4.7 Graphing Tangent and Cotangent 1. f(x) = tan(3x) 2. f(x) = - cot(3x)

1

-1-!

6

!

6

!

3

!

2

1

-1 !

6

!

3

!

2

2!

3

3. f(x) = cot(6x) 4. f(x) = - tan(6x)

1

-1 !

12

!

6

!

4

!

3

1

-1-!

12

!

12

!

6

!

4

5. f(x) = - cot( )1

4x 6. f(x) = tan( )14x

1

-1 2! 4! 6! 8!

1

-1 2!-2! 4! 6!

7. f(x) = - tan( )2

5x 8. f(x) = cot( )25x

1

-1-5!

4

5!

4

5!

2

15!

4

1

-1 5!

4

5!

2

5!15!

4

9. f(x) = tan( )32x 10. f(x) = -cot( )3

2x

1

-1-!

3

!

3

2!

3

!

1

-1 !

3

! 4!

3

2!

3

11. f(x) = cot( )4

3x 12. f(x) = -tan( )43x

1

-1 3!

8

3!

4

9!

8

3!

2

1

-1-3!

8

3!

8

3!

4

9!

8

Section 4.8 Inverse Trigonometric Functions

1. π6 2.

3π4 3. -

π4 4.

π2

5. π6 6. -

π6 7. 0 8.

π3

9. - π6 10. 0 11. -

π3 12.

π3

13. π4 14.

5π6 15.

π4 16. -

π4

17. π6 18.

π4 19. π 20.

2π3

21. - π3 22. -

π2 23.

π2 24.

π3

25. 0 26. - π3 27. -

π6 28. -

π3

29. π3 30. π 31.

5π6 32.

π2

33. - π2 34. -

π4 35. 0 36.

π3