Math 12 – Final Exam Review 1

Part One – Calculators are NOT PERMITTED for this part of the exam.

1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 ≤ θ ≤ 2π? 2

b) Draw these angles in standard position.

2. a) The cosine of angle θ is √3 What are the 2 possible values of θ in the domain 0 ≤ θ ≤ 2π? 2

b) Draw these angles in standard position.

Math 12 – Final Exam Review 2

3. The graphs of y = sinx and y = cosx are given. What are the amplitude, domain, period, range and

zeros for each graph?

y = sinx y = cosx

4. The graphs of y = sinx and y = cosx are given. Draw the transformed functions in the form

y = a sin b(x − h) + k or y = a cos b(x − h) + k on the same graphs.

Given: y = sinx Given: y = cosx

Draw: y = ½ sin (x − π) Draw: y = 3 cos ½ (x) + 1

Math 12 – Final Exam Review 3

5. Explain how changing the value of k affects the graphs of y = sin x + k and y = cos x + k.

6. Explain how changing the value of h affects the graphs of y = sin (x + h) and y = cos (x + h).

7. Explain how varying the value of b affects the graphs of y = sin bx and y = cos bx.

8. Explain how varying the value of a affects the graphs of y = a sin x and y = a cos x.

9. Determine the amplitude, domain, period, phase shift, range and zeros of the graph of the

trigonometric function y = ½ sin (x – π/4)

10. Determine the amplitude, domain, period, phase shift, range and zeros of the graph of the

trigonometric function y = cos 2(x) + 3

Math 12 – Final Exam Review 4

11. The graph of y = tanx is given. What are the asymptotes, domain, period, range and zeros?

12. Determine the values of a, b, h and k that correspond to the given graph, and write the equation of

the function in the form y = a sin b(x − h) + k

13. Determine the values of a, b, h and k that correspond to the given graph, and write the equation of

the function in the form y = a cos b(x − h) + k.

Math 12 – Final Exam Review 5

14. a) Sketch a graph of the exponential function y = 2x

Identify the domain, range, asymptotes and intercepts.

b) On the same graph, sketch the graph of y = -(2)2(x) + 1, and Identify the domain, range, asymptotes

and intercepts.

15. Sketch the graph of the logarithmic function y = log2 x

Identify the domain, range, asymptotes and intercepts.

Math 12 – Final Exam Review 6

16. The graph of y = √𝑥 is given. Sketch the graph of the function y = -2 √(𝑥 + 3) – 2 by applying

transformations to the graph. State the domain and range.

17. Determine if the graph of the rational function y = x2 + 5x – 6

x2 – 5x + 4

will have an asymptote or a point of discontinuity for its non-permissible values. What are they?

18. a) Sketch the graph of a function that is the sum of the two functions given in the graph below.

b) Sketch the graph of a function that is the difference of the two functions given in the graph

below.

Draw the sum of the functions here Draw the difference of the functions here

Math 12 – Final Exam Review 7

c) Sketch the graph of a function that is the product of the two functions given in the graph below.

d) Sketch the graph of a function that is the quotient of the two functions given in the graph below.

Draw the product of the functions here Draw the quotient of the functions here

19. Identify the graph of the function f(x) = 2x

x - 1

Math 12 – Final Exam Review 8

Part Two – Calculators are PERMITTED for this part of the exam.

1. Change from radians to degrees.

a) π b) 3.3 c) π d) 0.7

4

2. Change from degrees to radians.

a) 30˚ b) 123˚ c) 360˚ d) 240˚

3. Write a coterminal angle for the given angle.

a) 45˚ b) 188˚ c) 450˚ d) π

2

4. Draw the angles in standard position.

a) 45˚ b) 188˚

c) 450˚ d) π/2

Math 12 – Final Exam Review 9

5. Point P lies on the unit circle, forming an angle of 120˚. What are the exact coordinates (x,y) of point

P?

6. Solve the trigonometric equation -2cos2x + 1 = cotx sinx for the domain 0˚ ≤ x ≤ 360˚

7. Verify the trigonometric identity tan2x = 2tanx is true for x = π

1 – tan2x 3

8. Prove, algebraically, the trigonometric identity sin2x = 2sinxcosx is true for all values of x.

cos2x cos2x – sin2x

Math 12 – Final Exam Review 10

9. Sketch the graph of the inverse relation, given the graph of the relation here.

a) b)

Is the inverse of the relation a function? (yes/no)

10. Express the logarithmic expression as an exponential expression. Then find x.

a) logx625 = 4 b) log381 = x c) log749 = x d) log10x = 1

11. Express the exponential expression as a logarithmic expression.

a) y = 7x b) 27 = 3x c) 64 = 2x d) y = 106

Math 12 – Final Exam Review 11

12. Express as a single logarithm, using the laws of logarithms.

a) Logx2 + logx3 b) log12 – log4 c) log2 + logx – log(x+3) d) 5logx – 3logx

13. Solve.

a) 2log53 = log5(x + 1) b) log4(x2 + 1) – log46 = log45

14. Factor the polynomial.

a) f(x) = x3 - 8x2 - 20x b) 3x3 + 57x2 + 264x

Math 12 – Final Exam Review 12

15. Divide.

a) (x3 - 3x2 - 108x) ÷ (x - 12) b) (x4 - 15x3 + 11x2 + 375x - 900) ÷ (x - 12)

16. Which are factors of x3 – 2x2 – 31x – 28 ?

a) x + 4 b) x – 4 c) x – 7 d) x + 7

17. Write the equation of a function h(x), if h(x) = (f+g)(x)

a) f(x) = 4x + 6 and g(x) = 2x – 3

b) f(x) = ½ x – 7 and g(x) = -2x + 1

18. Write the equation of a function h(x), if h(x) = (f–g)(x)

a) f(x) = 6x + 2 and g(x) = 3x – 1

b) f(x) = ½ x + 3 and g(x) = -2x + 4

19. Write the equation of a function h(x), if h(x) = (f · g)(x)

a) f(x) = 4x – 2 and g(x) = ½ x + 8

b) f(x) = x + 3 and g(x) = -x2 + 5x – 6

20. Write the equation of a function that is the quotient of the two functions given.

a) f(x) = 2x + 6 and g(x) = x2 + 4x + 3

b) f(x) = x2 + 4x + 3 and g(x) = x + 1

Math 12 – Final Exam Review 13

21. Write the equation of a function f(g(x))

a) f(x) = 2x – 3 and g(x) = x2 – 5

b) f(x) = -1/2x and g(x) = (x + 4)2

22. In a certain city, the car number plate is formed by 3 digits from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9

followed by 3 letters from the alphabet.

a) How many number plates can be formed if neither the digits nor the letters are repeated?

b) How many number plates can be formed if the first number cannot be a 0?

c) How many number plates can be formed if repetition is allowed?

23. A school of 40 students needs to select 6 students to represent them at a speaking competition.

a) How many different teams are possible?

b) How many different teams are possible, if at least 3 students must be grade 12’s? (The school

has 13 grade 12’s)

Math 12 – Final Exam Review 14

24. Use the binomial theorem to expand (2a – 6)8

a) write it with coefficients in terms of nCr

b) write it by replacing the nCr terms with the numbers they represent.

25. If you were to expand (x + y)14

a) How many terms would be in the expanded expression?

b) what is the simplified 6th term in the expanded expression?