Waterloo-Oxford Mathematics Department



Review of Prerequisite Skills

The following questions can be found in Appendix A of the textbook.

Simplifying expressions Pg. 558 # 1 Simplify.

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

d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]

Factoring [pic] Pg. 549 # 1 Factor.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]

Factoring [pic] Pg. 549 # 1 Factor.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]

Factoring [pic] Pg. 548 # 1 Factor.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]

a. Solving quadratic equations by factoring Pg. 560-561 # 1 Solve by factoring.

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

Pg. 560-561 # 2 Solve by factoring.

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

Answers

c. Simplifying expressions Pg. 558 # 1:

d. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]

e. g) [pic] h) [pic] i) [pic]

Factoring [pic] Pg. 549 # 1:

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic]

Factoring [pic] Pg. 549 # 1:

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]

Factoring [pic] Pg. 548 # 1:

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

e) [pic] f) [pic]

f. Solving quadratic equations by factoring Pg. 560-561 # 1:

g. a) [pic] b) [pic] c) [pic] d) [pic]

h. Pg. 560-561 # 2

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

MCR3UI Content Review

Show all of your work.

1. Solve. a) [pic] b) [pic]

2. Which relation is NOT a function? Explain why it is not a function.

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

3. Which relation is NOT a function? Explain why it is not a function.

a) {(5,2),(3,1),(-6,7),(8,2)} b) {(3,1),(5,2),(8,-1),(5,7)}

c) {(3,7),(5,1),(-6,8),(7,2)} d) {(5,3),(2,-1),(-6,8),(8,5)}

4. Given [pic], determine [pic].

5. If [pic] and [pic], find: a) [pic] b) [pic]

6. State the domain and range of the function [pic].

7. Find [pic] for the function [pic]. Is the inverse a function?

Explain why/why not.

8. Simplify and state the restrictions for

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

9. Simplify a) [pic]

b) [pic]

10. For the function [pic], state the period, amplitude, phase

shift and vertical shift.

11. Solve to the nearest degree: [pic], for [pic].

12. Solve, for [pic], without a calculator.

a) [pic] b) [pic]

13. Find two angles, one positive and one negative, that are coterminal with:

a) [pic] b) [pic]

14. Prove the following identity: [pic]

15. From two different tracking stations, a weather balloon was spotted from two angles of elevation, [pic] and [pic], respectively. The altitude of the balloon is 28.5 km. How far apart are the tracking stations?

16. Two hot air balloons are moored directly over a level road. Based on the given diagram, how far apart are the balloons?

Answers:

1. a) 2 b) 2 2. d 3. b 4. -22 5. a) -9 b) -7 6. [pic]

7. [pic], not a function 8. a) [pic] b) [pic]

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

10. period=[pic], amplitude=2, phase shift left [pic], vertical shift up 4

11. [pic] 12. a) [pic] b) [pic] 13. a) [pic]

13. b) [pic] 15. 15 km 16. 2.4 km

Additional Textbook Questions

The following questions can be found in the textbook.

Pg. 18 # 1 Use interval notation to express the set of real values x described by each

inequality. Illustrate each interval on the real number line.

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

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

g) [pic] h) [pic]

Pg. 40 # 7abd Solve by factoring. Check your solutions.

a) [pic] b) [pic] d) [pic]

Pg. 40 # 8 Solve by factoring. Check your solutions.

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

Pg. 564 # 2abcf Solve using the quadratic formula. Express solutions as exact roots

and as approximate roots, to the nearest tenth.

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

Pg. 564 # 3abc Solve using the quadratic formula. Express solutions as exact roots.

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

Pg. 564 # 3def Solve using the quadratic formula. Express solutions as exact roots.

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

Answers

Pg. 18 # 1:

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

e) [pic] f) [pic] g) [pic] h) [pic]

Pg. 40 # 7abd:

a) 0, 8 b) -5, -2 d) 4, 5

Pg. 40 # 8:

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

Pg. 564 #2abcf:

a) [pic]; -0.9, 0.5 b) [pic]; -0.5, 1.2 c) [pic]; 1.3, 2.7

f) [pic]; -1.2, 0.4

Pg. 564 # 3abc:

a) no real roots b) no real roots c) no real roots

Pg. 564 # 3def:

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

Operations with Radicals

1. Simplify.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]

2. Express as entire radicals.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

3. Simplify.

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

e) [pic] f) [pic] g) [pic] h)[pic]

4. Simplify.

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

5. Simplify.

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

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

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

6. Simplify.

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

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

7. Simplify.

a) [pic] b) [pic]

Answers:

1. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] 2. a) [pic]

2. b) [pic] c) [pic] d) [pic] e) [pic] 3. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic]

3. h) [pic] 4. a) [pic] b) [pic] c) [pic] d) [pic] 5. a) [pic] b) [pic] c) [pic] d) [pic] 5. e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] 6. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

6. f) [pic] 7. a) [pic] b) [pic]

Working with Quadratic Equations

1. Determine a possible quadratic equation, in the form [pic],

with the given roots.

a) 2 and [pic] b) [pic] and [pic] c) [pic] and [pic] d) [pic] and [pic]

e) [pic] and [pic] f) [pic] and [pic] g) [pic] and [pic]

h) [pic] and [pic] i) [pic] and [pic]

2. Determine a possible quadratic equation, in the form [pic], with the given roots.

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

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

g) [pic] and [pic]

3. Solve. Check ONE of the solutions.

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

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

Answers:

1. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]

1. g) [pic] h) [pic] i) [pic]

2. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

2. f) [pic] g) [pic]

3. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

3. f) [pic]

Operations with Complex Numbers

4. Simplify.

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

e) [pic] f) [pic] g) [pic] h) [pic]

5. Evaluate.

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

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

6. Simplify.

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

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

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

7. Simplify. State answers in the form [pic].

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

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

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

8. Simplify.

a) [pic] b) [pic]

Answers:

1. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic]

2. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] 3. a) [pic] b) [pic]

3. c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic]

3. i) [pic] 4. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic]

4. g) [pic] h) [pic] i) [pic] 5. a) [pic] b) [pic]

Number Systems Recap

1. Simplify.

a) [pic] b) [pic]

2. Factor.

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

3. Solve. Check ONE answer for parts f,g and i.

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

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

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

j) [pic]

4. Use interval notation to express the set of real numbers, x, described by each inequality. Illustrate each interval on the real number line.

a) [pic] b) [pic]

5. Determine a possible quadratic equation in the form [pic] with the given roots.

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

d) [pic] and [pic] e) [pic] and [pic]

6. Simplify.

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

e) [pic] f) [pic] g) [pic] h) [pic]

i) [pic]

7. Simplify.

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

e) [pic] f) [pic]

8. Prove [pic].

Answers:

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

3. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic]

3. i) [pic] j) [pic] 4. a) [pic] b) [pic] 5. a) [pic] b) [pic]

5. c) [pic] d) [pic] e) [pic] 6. a) [pic] b) [pic] c) [pic] d) [pic]

6. e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] 7. a) [pic] b) [pic] c) [pic] d) [pic]

7. e) [pic] f) [pic]

Graphing Polynomial Functions in Factored Form

1. Given [pic].

a) State the end behaviour of [pic] as [pic] and [pic].

b) State the zeroes of [pic].

c) State the maximum number of turning points.

d) Sketch the graph of [pic]. Clearly label the x-intercepts.

2. Repeat question #1 for each polynomial below.

a) [pic] b) [pic]

c) [pic]

3. Sketch a possible graph of a polynomial function that satisfies each set of conditions.

a) degree four, positive leading coefficient, three zeroes, three turning points

b) degree four, negative leading coefficient, two zeroes, one turning point

c) degree four, positive leading coefficient, one zero, three turning points

d) degree three, negative leading coefficient, one zero, no turning point

e) degree three, positive leading coefficient, two zeroes, two turning points

4. Given [pic].

a) Sketch the graph of [pic].

b) Sketch another similar type of graph that uses the other zero as a turning point.

State a possible equation for this graph.

5. Sketch each function.

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

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

6. Sketch a possible graph of a polynomial function that satisfies each set of conditions:

a) degree three, negative leading coefficient, three distinct real roots

b) degree three, positive leading coefficient, two equal real roots and one real distinct root

c) degree three, positive leading coefficient, one real root and two complex roots

d) degree four, negative leading coefficient, two distinct real roots and two equal real roots

e) degree four, positive leading coefficient, two distinct pairs of equal real roots

f) degree four, positive leading coefficient, four complex roots

g) degree four, negative leading coefficient, three equal real roots and one distinct real root

h) degree five, negative leading coefficient, two distinct pairs of equal real roots and one

distinct real root

i) degree five, positive leading coefficient, two equal real roots, two complex roots and one

distinct real root

j) degree five, negative leading coefficient, two equal real roots and three distinct real roots

k) degree five, negative leading coefficient, three equal real roots and two complex roots

l) degree five, positive leading coefficient, three equal real roots and two distinct real roots

Polynomial Functions

1. Determine the equation of the cubic function, in standard form, with roots

[pic] and [pic] and passes through the point (-1, 15).

2. Determine the equation of the cubic function, in standard form, with roots

[pic] and [pic] which passes through the point (-3, -103).

3. Determine the equation of the quartic function, in standard form, with roots

[pic] and [pic] and passes through the point [pic].

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

5. Factor: a) [pic] b) [pic] c) [pic]

6. Sketch each function and label the x-intercepts.

a) [pic] b) [pic]

7. Sketch a possible graph of a polynomial function that satisfies each set of conditions:

a) degree three, negative leading coefficient, one real distinct root and two equal

real roots

b) degree four, positive leading coefficient, two distinct real roots and two complex

roots

c) degree five, positive leading coefficient, two complex roots, two equal real roots

and one distinct real root

8. Determine the equation of the family of polynomial functions given roots [pic] and [pic].

9. Determine the equation of the cubic function, in standard form, with roots [pic] and [pic] and a y-intercept of 72.

10. Factor: a) [pic] b) [pic]

Answers

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

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

5. c) [pic] 8. [pic] 9. [pic]

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

Factoring Sum/Difference of Cubes

1. Factor.

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

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

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

j) [pic] k) [pic] l) [pic]

m) [pic]

2. Factor.

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

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

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

Answers:

1. a) [pic] b) [pic] c) [pic]

1. d) [pic]

1. e) [pic] f) [pic] g) [pic] h) [pic]

1. i) [pic] j) [pic] k)[pic]

1. l) [pic] m) [pic]

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

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

2. h) [pic] i) [pic]

Solving Polynomial Inequalities

1. Solve. Graph each solution on a real number line.

a) [pic] b) [pic]

2. Solve. Graph each solution on a real number line.

a) [pic] b) [pic]

c) [pic] d) [pic]

e) [pic] f) [pic]

3. Find the solutions that satisfy both [pic] and [pic].

4. In Canada, hundreds of thousands of cubic metres of wood are harvested each year. The function [pic], [pic], models the volume harvested, in cubic metres, where the year 1993 corresponds to [pic]. In which years were less than [pic] harvested?

5. During a normal five-second respiratory cycle in which a person inhales and then exhales, the volume of air in a person’s lungs can be modeled by [pic], where volume, [pic], is in litres and [pic] is the time in seconds, [pic]. In this cycle, when is the volume of air in the lungs more than 0.3 L?

6. Solve [pic]. Find two other cubic polynomial inequalities with the same interval solutions. Describe how you found them and why your method works.

7. Solve [pic]. Graph the solution on a real number line.

8. Solve. Graph each solution on a real number line.

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

Answers:

1. a) [pic] b) [pic] 2. a) [pic]

2. b) [pic] c) [pic] d) [pic] e) [pic]

2. f) no real solution 3. [pic] 4. [pic] 5. [pic]

6. [pic], possible answers: [pic], [pic]

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

Transformations of [pic] and [pic]

1. Given the following base function and transformations:

i) State the function notation.

ii) State the function equation (in simplest form).

iii) State the transformations of the function equation (in simplest form).

a) [pic]; reflected in the x-axis, vertically stretched by a factor of [pic], horizontally

stretched by a factor of [pic], move left 3 units and up 4 units

b) [pic]; vertically stretched by a factor of 0.1, reflected in the y-axis, horizontally stretched by a factor of 0.4, move right 4 units and down 5 units

c) [pic]; reflected in the y-axis, reflected in the x-axis, horizontally stretched by

a factor of [pic], move right 6 units and down 1 unit, vertically stretched by a factor of [pic]

d) [pic]; move left 3 units and up 2 units, reflected in the x-axis, horizontally stretched by a factor of 3, vertically stretched by a factor of 27

2. Graph the following transformed functions and state the domain and range.

a) [pic]; reflected in the y-axis, horizontally stretched by a factor of 4, move down 2

units, vertically stretched by a factor of 32

b) [pic]; move right 5 units, move down 2 units, vertically stretched by a factor of ¾,

reflected in the x-axis

c) [pic]; horizontally stretched by a factor of2, move right 3 units and down 4 units, vertically stretched by a factor of 16, reflected in the x-axis, reflected in the y-axis.

d) [pic]; horizontally stretched by a factor of 2, move left 3 units and up 5 units,

reflected in the y-axis, vertically stretched by a factor of 2.

3. Given the base function and function notation,

i) State the function equation (in simplest form)

ii) State the transformations of the function equation (in simplest form).

iii) Graph and state the domain and range.

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

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

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

ANSWERS:

1. a) [pic] b) [pic]

[pic] [pic]

- reflect in the x-axis - vertically stretched by a factor of [pic]

- vertically stretched by a factor of 10 - move right 4 units, down 5 units

- move left 3 units, up 4 units

c) [pic] d) [pic]

[pic] [pic]

- vertically stretched by a factor of 6 - reflect in the x-axis

- move right 6 units, down 1 unit - vertically stretched by a factor of [pic]

- move left 3 units, up 2 units

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

[pic] [pic] [pic] [pic]

- reflect in the x-axis - same as transformations given in question

- vertically stretched by a factor of 0.5

- move down 2 units

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

[pic] [pic] [pic] [pic]

- vertically stretched by a factor of 2 - vertically stretched by a factor of [pic]

- move right 3 units, down 4 units - move left 3 units, up 5 units

3. a) Simplest Form: [pic] [pic] [pic]

reflect in the x-axis, move left 3 units, up 4 units

b) Simplest Form: [pic] [pic] [pic]

vertically stretched by a factor of ½, move right 3 units, move up 1 unit

c) Simplest Form: [pic] [pic] [pic]

reflect in the x-axis, vertically stretched by a factor of 2, move left 1 unit, move down 4 units.

d) Simplest Form: [pic] [pic] [pic]

- reflect in the x-axis, vertically stretched by a factor of [pic], move left 4 units, down 1 units

e) Simplest Form: [pic] [pic] [pic]

- reflect in the x-axis, vertically stretched by a factor of [pic], move right 6 units, down 1 unit

f) Simplest Form: [pic] [pic] [pic]

- vertically stretched by a factor of [pic], move right 4 units, down 1 unit

Exponential Expressions, Equations and Functions

1. Evaluate without a calculator.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]

k) [pic] l) [pic] m) [pic] n) [pic] o) [pic]

p) [pic] q) [pic] r) [pic] s) [pic] t) [pic]

2. Solve.

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

3. Find x and y if [pic] and [pic].

4. Describe the transformations of each function below. Rewrite the equation as necessary.

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

e) [pic] f) [pic] g)[pic] h) [pic]

i) [pic] j)[pic] k) [pic] l) [pic]

b) Graph each function in #4 using transformations. Clearly label the asymptote.

c) State the domain and range of each function in #4.

Answers

1. a)[pic] b)[pic] c)[pic] d)[pic] e)[pic] f)[pic] g)[pic] h)[pic] i)[pic] j)[pic] k)[pic] l)[pic] m)[pic] n)[pic] o)[pic] p)[pic] q)[pic] r)[pic] s)[pic] t)[pic]

2. a) 5 b) -1 c) -1 d) 1 3. x = -17, y = 2

Logarithmic Functions, Equations and Expressions

1. Describe the transformations of each function below. Rewrite the equation as necessary.

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

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

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

j) [pic] k) [pic] l) [pic]

m) [pic] n) [pic] o) [pic]

p) [pic] q) [pic] r) [pic]

2. Graph each function in #1 using transformations. Clearly label the equation

of the asymptote.

3. State the domain and the range of each function in #1.

4. Evaluate without a calculator.

a) [pic] b) [pic] c) [pic] d) [pic] e) [pic]

f) [pic] g) [pic] h) [pic] i) [pic] j) [pic]

k) [pic] l) [pic] m) [pic]

Answers

4. a) [pic] b) [pic] c) [pic] d) [pic] e) [pic] f) [pic] g) [pic] h) [pic] i) [pic] j) [pic] k) [pic] l) [pic] m) [pic]

Additional Unit 4 Test Review

1. Evaluate each of the following, without a calculator. (except for part g)

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

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

g) [pic]

2. Solve, accurate to four decimal places. Calculators permitted.

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

3. Evaluate without a calculator. (except for part a)

a) [pic] (accurate to four decimal places) b) [pic] c) [pic]

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

h) [pic] i) [pic] j) [pic]

Answers:

1. a) 3 b) -4 c) 5 d) [pic] e) 4 f) 2 g) 4 2. a) 2.6486 b) 4.8451

2. c) -1.5979 3. a) 2.0206 b) 1 c) 0 d) 5 e) 67 f) [pic] g) [pic] h) [pic] i) 9 j) 16

Sketching Rational Functions

1. Sketch the following functions. Label the asymptotes. State positive intervals, negative intervals, increasing intervals and decreasing intervals.

a) [pic] b) [pic]

c) [pic] d) [pic]

2. Sketch the following reciprocal functions. Label the asymptotes.

a) [pic] b) [pic]

c) [pic] d) [pic]

3. Create a function that has a graph with the given features:

a) a vertical asymptote at x = 1 and a horizontal asymptote at y = 0.

b) a vertical asymptote at x = -3 and x = 5 and a horizontal asymptote at y = -2

Find the constants a and b that guarantee that the graph of the function defined by [pic] will have vertical asymptotes at [pic] and [pic] and a horizontal asymptote

at y = -2.

The model for the percentage y of a drug in the bloodstream, x hours after it is taken orally, is [pic].

a) What is the domain of x in this context?

b) What is the percentage (nearest tenth) of the drug is in the person’s bloodstream after

i) one hour ii) 2 hours iii) four hours

iv) twelve hours v) 24 hours

c) What are the key features of this graph?

d) Sketch the function.

e) Describe what happens to the concentration of the drug over 24 consecutive hours. Does the model seem reasonable?

ANSWERS:

|1. |[pic] |[pic] |

| | | |

| |positive: [pic], negative: [pic], |positive: [pic],negative:[pic] |

| |increasing: none, decreasing: [pic] |increasing: none, decreasing:[pic] |

| |[pic] | |

| | |[pic] |

| | | |

| |positive: [pic], negative: [pic], increasing: [pic], decreasing: |positive: [pic],negative: [pic], |

| |none |increasing: [pic], decreasing: none |

|2. |a) [pic] |b) [pic] |

| | | |

| |[pic] |[pic] |

| | | |

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

| | | |

| |[pic] |[pic] |

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

4. a = -50, b = 25

5. a) D = [pic]

b) i) 2.3% ii) 2.3 % iii) 1.6% iv) 0.6% v) 0.3%

c) horizontal asymptote y = 0; no vertical asymptote

|d) |e) The function increases to a |

|[pic] |maximum at (1.4, 2.5). |

| |The model seems reasonable. |

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D1

D2

D3

D4

D5

D6

D7

D8

D9

D10

D11

D12

D13

D14

D15

D16

D17

D18

D19

b) b)

a) b)

x = 3

y = 0

x = -2

y = -0.5

x = -2

c) d)

y = 2

x = 5

y = -2

y = 0

x = -2

x = 6

y = 0

x = 3

x = -4

x = 0

x = 3

y = 0

D20

x = -0.5

x = -4

x = 1

y = 0

y = 0

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