MHF 4UI Unit 6



Applications

Round all final answers correctly to one decimal unless the question says otherwise.

1. At the Luanda port, the depth of the water h metres at time t hours during a certain day is given by the formula [pic].

a) Determine the amplitude, period, phase shift and vertical shift.

b) Use the information to sketch the function over a 24 hour interval.

c) Determine the maximum water depth and the time(s) it occurs.

d) Determine the depth of the water at 6:30 a.m. and 7:30 p.m.

2. At Yourtown, the time of the sunrise on the nth day of the year is given by the formula. [pic].

a) Determine the time, to the nearest minute, that the sun rises on October 20 (day 293).

b) What is the earliest sunrise and on what day does it occur? Round the time to the nearest minute.

3. Given the function [pic], where h is the height of an object in metres relative to the ground at any time t seconds.

a) Determine the period of the function.

b) Determine the minimum value and when it occurs.

c) Determine the height after 15 seconds.

d) Determine all values of t such that the object is 6 m above the ground.

4. A Ferris wheel with a radius of 20 m rotates once every 40 s. Passengers get on at the lowest point which is 1 m above the ground.

a) Draw a graph showing your height above the ground during the first two cycles.

b) Write an equation which expresses your height as a function of time on the ride.

c) Calculate your height above the ground after 15 s.

d) At what times will the rider be 30 m above the ground?

e) At what times in the first rotation will the rider be 25 m or higher above the ground?

NOTE: An algebraic solution is required but refer to your graph from

part a) to interpret your answers.

5. Tidal forces are the greatest when the Earth, the sun, and the moon are in line. When this occurs at the Annapolis Tidal Generating Station in Nova Scotia, the water has a maximum depth of 9.6 m at 4:30 a.m. and a minimum depth of 0.4 m approximately 6 hours later.

a) Write an equation for the depth of the water at any time t.

b) Calculate the depth of the water at 9:30 a.m. and 6:45 p.m.

c) At what times during the day will the water be at least 9 m deep? Round your times to the nearest minute. A sketch may be helpful in interpreting your answers.

6. The pedals of a bicycle are mounted on a bracket whose centre is 30 cm above the ground. Each pedal is 16 cm from the bracket. The bicycle is pedalled at 6 cycles per minute. Assume that you start pedalling with the pedal at the highest position at t = 0.

a) Draw a graph showing the height of the pedal above the ground during the first minute.

b) Determine the amplitude, period, and vertical shift.

c) Write an equation to represent the height of the pedal as a function of time.

d) Determine the height of the pedal after 25 seconds.

e) At what times in the first 30 seconds will the pedal be 15 cm above the ground? Round your times to the nearest tenth of a second.

f) At what times will the pedal be at least 15 cm above the ground?

Answers

1. a) amp = 1.5, period = 12, ps = 4, vs = 3.1 c) max depth = 4.6 m, [pic]

1. d) 4.5 m, 4.5 m 2. a) 5:28 a.m. b) 4:31 a.m. on day 354 (December 20)

3. a) period = 12.4 b) min = 1.8 m when [pic] c) 6.4 m

3. d) 3.1 s and 12.3 s; in general, [pic]

4. b) if ps = 0, [pic] OR [pic] c) 35.1 m

4. d) 13.0 s and 27.0 s; in general, [pic]

4. e) between 11.3 s and 28.7 s

5. a) if ps = 4.5, [pic]; if ps = 1.5, [pic]

5. b) 1.0 m, 6.8 m

5. c) between 3:31 a.m. and 5:29 a.m. and between 3:31 p.m. and 5:29 p.m.

6. b) amp = 16, period = 10, vs = 30 c) if ps = 0, [pic] d) 14 cm

6. e) 4.4 s, 5.6 s, 14.4 s, 15.6 s, 24.4 s, 25.6 s

6. f) between: 0 s and 4.4 s & 5.6 s and 14.4 s & 15.6 s and 24.4 s

& 25.6 s and 30 s

UNIT #6 Review

1. Convert each degree to radian measure in terms of (.

|a) 45° |b) 300° |c) -210° | |

2. Convert each radian measure to degrees. If the answer is not exact, round correctly to one decimal place.

|a) [pic]rad |b) [pic] rad | | |

| | |c) 6 rad |d) 3.5( rad |

3. Find the length of the arc of a circle with radius 40 cm and central angle 50°. State the exact answer and the answer accurate to 3 decimal places. (The central angle is the angle subtended at the centre of the circle by the arc.)

4. In a circle, the arc length is 9 mm and the angle subtended at the centre is [pic]. What is the exact area of the circle?

5. Determine the related acute angle associated with each of the following standard position angles. Provide a clearly labelled sketch.

|a) [pic] |b) [pic] |c) [pic] |d) 4.25 |

6. ( is in standard position with its terminal arm in the quadrant 4, sin( = [pic], and

0 < ( < 2(. Provide a clearly labelled sketch. Determine the exact value of cos( and tan(.

7. For each of the following, (i) sketch the standard position angle

(ii) determine the related acute angle

(iii) determine the exact value of the specified trigonometric ratio. Do not use a calculator.

|a) tan [pic] |b) sin [pic] |c) cos[pic] |d) tan [pic] |

8. Solve each of the following equations to two decimal places, [pic].

a) [pic] b) [pic] c) [pic]

9. Solve each of the following equations. Give exact answers only, [pic].

a) [pic] b) [pic] c) [pic]

10. Solve each of the following equations.

a) [pic], [pic], correct to two decimal places

b) [pic], [pic], exact answers.

11. Solve [pic]for x, [pic]. Give exact answers. Hint – factor.

12. On the same grid, sketch each of the following pairs of graphs. Clearly label all

asymptotes and zeroes. Review the features of the various trigonometric functions.

a) [pic][pic], [pic]

b) [pic][pic]

c) [pic]

For each of the following functions,

i) determine the amplitude, period, phase shift and vertical shift;

ii) graph at least one complete period; and

iii) state the domain and range.

a) [pic] b) [pic]

c) [pic] d) [pic]

14. Determine the equation of a cosine function with maximum value 5, minimum value -9,

period [pic] and phase shift [pic].

15. Determine the equation of a sine function with a minimum at ([pic],0) and a maximum

at [pic].

For each of the following trigonometric equations:

( Write the amplitude, period, phase shift & vertical shift.

( Write the equation of the function.

16. Cosine Function Equation: ________________________________

Amplitude = _____

Period = _____

Phase Shift = _____

Vertical Shift. = _____

17. Sine Function Equation: ___________________________________

Amplitude = _____

Period = _____

Phase Shift = _____

Vertical Shift. = _____

18. Given the function,[pic]:

a) Determine the domain and range.

b) Determine the amplitude, period, phase shift and vertical shift.

c) Determine d, accurate to one decimal place when t = 11.

d) Determine all values of t such that d is 8. State the exact answers.

19. A Ferris wheel with a diameter of 36 m rotates three times every two minutes. Passengers get on at the lowest point which is 6 m above the ground.

a) Draw a graph showing the height of the rider above the ground during the first two minutes.

b) Write an equation which expresses your height as a function of time on the ride.

c) Calculate the exact height above the ground after 25 s.

d) At what times in the first two rotations will the rider be 10 m or less above the ground?

Answers

1. a) [pic] b) [pic] c) [pic] 2. a) 150( b) 405( c) 343.8( d) 630( 3. [pic]cm, 34.907 cm 4. [pic]mm2

5. a) [pic] b) [pic] c) [pic] d) 1.11 6. [pic] 7. a) 1 b) [pic] c) [pic] d) undefined 8. a) 3.28, 6.15

8. b) 1.32, 4.96 c) 2.82, 5.96 9. a) [pic] b) [pic] c) [pic] 10. a) 1.25, 2.94 10. b) [pic]

11. [pic] 12. refer to course notes 13. a) i) a=2, period=[pic], ps=[pic], vs=0

13. a) iii) D=[pic], R=[pic] b) i) a=1, period=[pic], ps=[pic], vs=0 iii) D=[pic], R= [pic]

13. c) i) a=3, period=[pic], ps=[pic], vs=-2 iii) D=[pic], R=[pic] d) i)a=0.5, period=[pic], ps=[pic], vs = 2.5

13. d) iii) D=[pic], R=[pic] 14. [pic]

15. [pic] OR [pic]

16. a=1.5, period=[pic], ps=[pic], vs=1, [pic] 17. a=2, period=[pic], ps=0, vs=2, [pic]

18. a) D=[pic], R=[pic] b) a=4, period=20, ps=3, vs=6 c) 8.4 d) [pic]

19. b) a=18, period=40, vs=24, if ps=0 then [pic] c) [pic]

19. d) between: 0 and 4.3 seconds, 35.7 and 44.3 seconds, and 75.7 and 80 seconds

Trigonometric Identities

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] |[pic] |

|[pic] | |

Trigonometric Identities

1. Prove the following trigonometric identities.

|a) [pic] |b) [pic] |

|c) [pic] |d) [pic] |

|e) [pic] |f) [pic] |

|g) [pic] |h) [pic] |

|i) [pic] | |

2. Prove: (a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

3. Prove: (a) [pic] (b) [pic]

4. Prove: (a) [pic] (b) [pic]

(c) [pic]

5. Prove: (a) [pic] (b) [pic]

6. Prove: (a) [pic] (b)[pic]

Cofunction Identities

1. Write each of the following in terms of the cofunction identity.

a) [pic] b) [pic] c) [pic]

d) [pic] e) [pic] f) [pic]

2. Write [pic] in terms of secant. Recall that cosecant is the reciprocal of the sine

function and secant is the reciprocal of the cosine function.

Answers: 1. a) [pic] b) [pic] c) [pic] d) [pic]

e) [pic] f) [pic] 2. [pic]

More Trigonometric Identities

3. Prove the following trigonometric identities.

a) [pic]

b) [pic]

c) [pic]

4. Prove the following trigonometric identities.

a) [pic] b) [pic]

5. CHALLENGE:

Prove: [pic].

You may use your results from page 503 #8.

Equations and Identities

Solve the following equations for x, [pic]. State exact answers where possible; otherwise, state answers accurate to two decimal places.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

Answers:

1. [pic] 2. [pic] 3. [pic]

4. [pic] 5. [pic]

6. [pic] 7. [pic] 8. [pic]

Prove the following trigonometric identities.

9. [pic] 10. [pic]

11. [pic] 12. [pic]

13. [pic] 14. [pic]

15. [pic] 16. [pic]

17. [pic]

Trigonometric Equations Involving Different Functions

Solve the following equations for x, [pic] State the exact answers where

possible. Otherwise round correctly to three decimal places.

1. [pic] 2. [pic]

3. [pic] 4. [pic]

5. [pic] 6. [pic]

7. [pic] 8. [pic]

9. Find the exact points of intersection of [pic]

on the domain [pic]

10. We know that [pic] is an identity and hence is true for all values of x. However, [pic]is not an identity. Over the interval [pic], determine the values of x for which the equation is satisfied. You may have to check your answers when done.

[HINT: squaring both sides carefully may help.]

Answers

1. [pic] 2. [pic] 3. [pic] 4. [pic]

5. [pic] 6. 2.944, 6.086 7. [pic]

8. [pic]

9. [pic] 10. [pic]

Key Features of Functions

1. Sketch the following polynomial functions and identify the key features: the x-intercept(s), y-intercept, degree of the polynomial function , number of zeroes, end behaviours and finite differences.

a) [pic] b) [pic]

2. Sketch the following exponential and logarithmic functions and identify the key features: domain, range, equation of the asymptote, end behaviours, increasing/decreasing function, x-intercept(s), y-intercept and finite differences.

a) [pic] b) [pic]

3. Sketch [pic] and identify the key features: vertical and horizontal asymptotes, domain, range, x-intercept(s), y-intercept, increasing intervals, decreasing intervals, positive intervals and negative intervals.

4. Sketch [pic] and identify the key features: vertical and horizontal asymptotes, domain, range, x-intercept(s), y-intercept, increasing intervals, decreasing intervals, positive intervals and negative intervals.

5. Sketch [pic] and identify the key features: vertical, horizontal and linear oblique asymptotes, x-intercept(s) and y-intercept.

Answers

1. a) x-ints -3,1,4; y-int -12; degree 3; 3 zeroes

[pic]

The difference table shows third differences of y

have a constant value of -6 for constant (x.

b) x-ints -1,2, [pic], [pic]; y-int 8; degree 4;

4 zeroes

[pic]

The difference table shows fourth differences of y

have a constant value of 24 for constant (x.

2. a) vertical asymptote x=-1; no y-int; x-int -1.6

[pic]; [pic]

increasing: none; decreasing: [pic]

[pic]

The difference table shows a constant ratio of 2

between first differences of x values for

constant (y.

b) horizontal asymptote y=-5; y-int [pic]; x-int 4.7

[pic]; [pic]

increasing: [pic]; decreasing: none

[pic]

The difference table shows a constant ratio of 2

between first differences of y values for

constant (x.

3. vertical asymptote x=3; horizontal asymptote y=0

y-int [pic]; no x-int; [pic]; [pic]

4. vertical asymptote x=2; horizontal asymptote [pic]

y-int [pic]; x-int 3; [pic]; [pic]

positive: [pic]; negative: [pic];

increasing: none; decreasing: [pic]

5. vertical asymptote x=3; no horizontal asymptote;

linear oblique asymptote y=x+9; y-int [pic]; x-int -4,-2

More Working with Intercepts

1. For each of the following:

a) State the domain

b) State the range.

c) Determine the x–intercepts. State exact answers, then accurate to two decimal places.

d) Determine the exact value of the y–intercept.

i) [pic]

ii) [pic]

iii) [pic]

2. For each of the following:

a) Determine the amplitude, vertical shift, period and phase shift.

b) Determine the maximum, minimum, domain and range.

c) Determine the exact value of the y–intercept, if possible; otherwise, state

answers accurate to two decimal places.

d) Determine the exact value of the x-intercept(s) in one period, if possible;

otherwise, state answers accurate to two decimal places.

e) Determine the exact value of all zeroes, if possible; otherwise, state

answers accurate to two decimal places.

i) [pic]

ii) [pic]

iii) [pic]

iv) [pic]

Answers:

1. i) Domain = [pic], Range = [pic], x-int = [pic] (= 2.20), y-int = [pic]

ii) Domain = [pic], Range = [pic], x-int = [pic] (= -0.18), y-int = -2

iii) Domain = [pic], Range = [pic], x-int = [pic], y-int = [pic]

2. i) a) amp = 5, vs = 4, period = [pic], ps = 0

b) maximum = 9, minimum = -1

Domain = [pic], Range = [pic]

c) y – intercept = 4 d) x – intercepts = [pic]

e) zeroes [pic]

2. ii) a) amp = 2, vs = 1, period = [pic], ps = [pic]

b) maximum = 3, minimum = -1

Domain = [pic], Range = [pic]

c) y – intercept = [pic] d) x – intercepts = [pic]

e) zeroes [pic]

2. iii) a) amp = 4, vs = 6, period = [pic], ps = [pic]

b) maximum = 10, minimum = 2

Domain = [pic], Range = [pic]

c) y – intercept = 4 d) x – intercepts = none e) zeroes: none

2. iv) a) amp = 3, vs = -3, period = [pic], ps = [pic]

b) maximum = 0, minimum = -6

Domain = [pic], Range = [pic]

c) y – intercept = -4.03 d) x – intercept = [pic]

e) zeroes [pic]

Sum and Differences of Functions

1. Given the functions [pic] and [pic],

|i) i) State the domain of [pic]. |iii) Find [pic] and state the domain. |

|ii) State the domain of [pic]. |iv) Find [pic] and state the domain. |

| |v) Find [pic] and state the domain. |

a) [pic] and [pic] b) [pic] and [pic]

c) [pic] and [pic] d) [pic] and [pic]

2. Given the functions [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain.

d) Find [pic].

e) Determine [pic] and state the domain.

f) Find [pic].

3. Given the functions [pic] and [pic],

a) State the domain of [pic].

b) State the domain of [pic].

c) Determine [pic] and state the domain.

d) Find [pic].

e) Determine [pic] and state the domain.

f) Find [pic].

g) Determine [pic] and state the domain.

h) Find the exact value of [pic]. Then, find accurate to two decimal places.

4. Given the functions [pic] and [pic],

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain.

d) Find the exact value of [pic].

e) Determine [pic] and state the domain.

f) Find the exact value of [pic].

g) Determine [pic] and state the domain.

h) Find the exact value of [pic].

ANSWERS:

1. a) [pic]; [pic]; [pic]; [pic]

[pic]; [pic]; [pic]; [pic]

b) [pic]; [pic]; [pic];

[pic]; [pic]; [pic];

[pic]; [pic]

c) [pic]; [pic];

[pic]; [pic]; [pic];

[pic]; [pic]; [pic]

d) [pic]; [pic]

[pic]; [pic]

[pic]; [pic]

[pic]; [pic]

2. a) [pic] b) [pic] c) [pic]; [pic]

d) [pic] e) [pic]; [pic] f) [pic]

3. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic]

e) [pic]; [pic] f) [pic]

g) [pic]; [pic] h) [pic]

4. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic]

e) [pic]; [pic] f) [pic]

g) [pic]; [pic] h) [pic]

Products and Quotients of Functions

1. Given the functions [pic] and [pic],

|i) i) State the domain of [pic]. | iv) Find [pic] and state the domain. |

|ii) State the domain of [pic]. |v) Find [pic] and state the domain. |

|iii) Find [pic] and state the domain. | |

a) [pic] and [pic]

b) [pic] and [pic]

c) [pic] and [pic]

2. Given the functions [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find [pic].

e) Determine [pic] and state the domain. f) Find [pic].

g) Determine [pic] and state the domain. h) Find [pic].

3. Given the functions [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain.

d) Find the exact value of [pic]. Then, state accurate to two decimal places.

e) Determine [pic] and state the domain.

f) Find the exact value of [pic]. Then, state accurate to two decimal places.

g) Determine [pic] and state the domain.

h) Find the exact value of [pic]. Then, state accurate to two decimal places.

4. Given the functions [pic] and [pic],

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find the exact value of [pic].

e) Determine [pic] and state the domain. f) Find the exact value of [pic].

g) Determine [pic] and state the domain. h) Find the exact value of [pic].

ANSWERS:

1. a) [pic]; [pic]; [pic]; [pic]

[pic] (hole at x = -2); [pic]

[pic] (hole at x = -2); [pic]

b) [pic]; [pic];

[pic]; [pic]

[pic] (hole at x = -1 is outside [pic]) ; [pic]

[pic] (hole at x = -1 is outside [pic]); [pic]

c) [pic]; [pic]

[pic] (hole at x = -6); [pic]

[pic] hole at x = -2; [pic]

[pic] hole at x = -2 ; [pic]

2. a) [pic] b) [pic] c) [pic]; [pic]

d) [pic] e) [pic]; [pic] f) [pic]

g) [pic] [pic] h) [pic]

3. a) [pic] b) [pic]

c) [pic]; [pic]

d) [pic] e) [pic]; [pic]

f) [pic] g) [pic]; [pic]

h) [pic]

4. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic]

e) [pic]; [pic] f) [pic]

g) [pic]; [pic] h) [pic]

UNIT #8 REVIEW: Combination of Functions

1. Sketch the following functions:

i) [pic][pic] ii) [pic]

Identify the key properties listed below. Then, sketch the function.

a) x-intercept(s) and state number of zeros b) y-intercept

c) degree d) end behaviours e) domain

f) If you were to complete a finite differences chart, describe the pattern that you would see.

2. Sketch the following functions:

i) [pic] ii) [pic]

Identify the key properties.

a) x-intercept b) y-intercept c) increasing/decreasing function

d) vertical asymptote e) horizontal asymptote f) end behaviours

g) domain and range h) If you were to complete a finite differences

chart, describe the pattern that you would see.

3. Sketch [pic].

Identify the key properties.

a) vertical asymptote(s) b) horizontal asymptote(s) c) linear oblique asymptote

d) hole(s) e) domain and range f) x-intercept g) y-intercept

h) positive and negative intervals i) increasing and decreasing intervals

4. Sketch the following functions:

i) [pic] ii) [pic]

Determine the following:

a) amplitude, vertical shift, phase shift and period

b) minimum value, maximum value, domain and range

c) y-intercept (state exact value if possible; otherwise, state answers accurate to 2 decimal places)

d) Determine the x-intercept(s) in one period (state the exact value where possible; otherwise, state answers accurate to 2 decimal places).

e) Determine the value of all zeroes.

5. For [pic] and [pic], determine:

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find [pic] (exact answer).

e) Determine [pic] and state the domain. f) Find [pic] (round to 2 decimal places).

6. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find [pic] (round to 2 decimal places).

e) Determine [pic] and state the domain. f) Find [pic] (round to 2 decimal places).

7. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find [pic].

e) Determine [pic]and state the domain. f) Find [pic].

8. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) Find [pic] (exact answer).

e) Determine [pic] and state the domain. f) Find [pic] (exact answer).

g) Determine [pic] and state the domain. h) Find [pic] (exact answer).

9. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) [pic]

e) Determine [pic] and state the domain. f) [pic]

10. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) [pic]

e) Determine [pic] and state the domain. f) [pic]

11. For [pic] and [pic]

a) State the domain of [pic]. b) State the domain of [pic].

c) Determine [pic] and state the domain. d) [pic]

e) Determine [pic] and state the domain. f) [pic]

12. For the given function, determine functions f and g such that [pic].

a) [pic] b) [pic] c) [pic]

ANSWERS:

|1 i) |[pic] |a) -2, 1, 5 (3 zeros) b) 20 c) degree = 4 |

| | |d) [pic] e) [pic] |

| | |f) QUARTIC: For constant (x, the 4th differences of y |

| | |are constant. |

| | | |

|1 ii) |[pic] |a) -2, -1, 3 (3 zeros) b) -6 c) degree = 3 |

| | |d) [pic] |

| | |e) [pic] f) CUBIC – For constant (x, the 3rd differences |

| | |of y are constant. |

|2 i) |[pic] |a) 1 b) 6 c) decreasing d) none |

| | |e) y = -6 |

| | |f) [pic] |

| | |g) [pic]; [pic] |

| | |h) EXPONENTIAL – For constant (x, there is a |

| | |constant ratio between the 1st differences in y. |

|2 ii) |[pic] |a) -0.37 b) -2 c) decreasing d) x = -1 |

| | |e) none |

| | |f) [pic] |

| | |g) [pic]; [pic] |

| | |h) LOGARITHMIC – For constant (y, there is a |

| | |constant ratio between the 1st differences in x. |

| | | |

|3. | |a) x = -1 b) y = 2 c) none |

| | |d) hole at x = -5 (y-value of hole is 4) |

| | |e) [pic];[pic] |

| | |f) 3 g) -6 |

| | |h) positive: [pic]; |

| | |negative: [pic] |

| | |increasing: [pic]; |

| | |decreasing: none |

|4 i) |[pic] |a) amplitude = 2, vs = 1, ps = [pic], period = [pic] |

| | | |

| | |b) min = -1, max = 3, |

| | |[pic], [pic] |

| | |c) [pic] (2.73) d) One Period: [pic] |

| | | |

| | |e) All zeros: [pic] |

|4 ii) |[pic] |a) amplitude = 3, vs = -2, ps = -[pic], period = [pic] |

| | | |

| | |b) min = -5, max = 1, |

| | |[pic], [pic] |

| | |c) [pic] (-3.5) d) One Period: [pic] |

| | | |

| | |e) All zeros: [pic] |

5. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic]

e) [pic]; [pic] f) [pic] = -2.13

6. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic] = 3.26

e) [pic]; [pic] f) [pic] = -1.14

7. a) [pic] b) [pic] c) [pic]; [pic] d) [pic]

e) [pic], hole at x = 3; [pic] f) [pic]

8. a) [pic] b) [pic]

c) [pic]; [pic] d) [pic]

e) [pic]; [pic] f) [pic]

g) [pic]; [pic] h) [pic]

9. a) [pic] b) [pic] c) [pic]; [pic]

d) [pic] e) [pic] ; [pic] f) [pic]

10. a) [pic] b) [pic] c) [pic]; [pic]

d) [pic] e) [pic]; [pic] f) [pic]

11. a) [pic] b)[pic] c) [pic]; [pic]

d) [pic] e) [pic]; [pic] f) [pic]

12. a) [pic], [pic] b) [pic], [pic] c) [pic], [pic]

-----------------------

D41

D42

D43

D44

[pic]

D45

D46

D47

D48

D49

D50

D51

D52

x = -1

y = -5

y = 0

x = 3

y = [pic]

x = 2

x = 3

y = x+9

D53

D54

D55

D56

.

D57

D58

.

.

.

D59

D60

D61

y = -6

x = -1

Hole at

x = -5

o

x = -1

y = 2

D62

.

.

.

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