Final Exam Practice PDF

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practice test for the final exam, advanced functions...

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MHF4U Practice Final Exam Part A – Multiple Choice [K/U – 20 marks] 1. Which of the following is a polynomial function? a. y = sin x c. y = 3x b. y = cos x d. y = x3 2. What is the degree and lead coefficient of 𝑓(𝑥) = −𝑥 + 5𝑥 2 + 6𝑥3 + 10? a. degree 1 with a lead coefficient of −1 c. degree 3 with a lead coefficient of −6 d. degree 6 with a lead coefficient of −1

b. degree 3 with a lead coefficient of 6

3. The least possible degree of the polynomial function represented by the graph shown is

a. b.

2 3

c. d.

4. For a polynomial P(x), if P

= 0, then which of the following must be a factor of P(x)?

a. b.

4 5

3x + 5

c.

5x + 3

d.

5x – 3

5. What are the x-intercepts of the graph of a. b.

–4, 5 –7, 3

c. d.

6. Starting on the positive x-axis and moving a.

3.14 m

b.

7. Determine the exact value of a. b.

0 –1

360 m

? 4, –5 7, –3 of the way around a circle that has radius 3 m, how far would one travel? c.

6.28 m

. c. d.

1 undefined

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d.

0.69 m

8. Determine the average rate of change of the function a.

c.

b.

d.

9. An equivalent trigonometric expression for a. b.

c. d.

from

to

is none of the above

10. Which of the following functions has the longest period? a. c. b.

d.

11. Evaluate a. b.

. c. d.

–12 8

12. Solve the equation

.

a.

c.

b.

d.

13. The function

6 –4

is

a. compressed vertically by a factor of

c. translated down 3 units

b. stretched horizontally by a factor of

d. translated right 3 units

14. Evaluate: a. b.

2 4

c. d.

15. State the law of logarithms used to rewrite a. Difference law of logarithms b.

Product law of logarithms

6 8

c.

. Power law of logarithms

d.

Quotient law of logarithms

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.

16. Given the functions

and

a.

c.

b.

d.

17. Given the functions a. b.

and

19. If a. b.

.

, determine the domain of the combined function c. d.

18. Given the functions a. 5 b. 8

, determine an equation for the combined function

and c. d.

.

cannot be determined , determine the value of

.

13 25

is a non-constant linear function and is a quadratic function, then what type of function is linear c. cubic quadratic d. cannot be determined for sure

20. Given the functions

and

, determine the range of the combined function

a.

c.

b.

d.

?

.

Part B – Thinking and Investigation [TI – 24 marks] **Show ALL of your work. 1. Solve the following inequalities using an algebraic method. ✓✓ & ✓✓✓ a) 𝑥 (𝑥 − 2)(𝑥 − 1)(𝑥 + 1) ≤ 0

2. Determine an equation in the form intercept of

b)

for a function with a vertical asymptote at 𝑥 = −2 and a y-

1

. ✓✓

8

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is an identity. ✓✓✓

3. Prove that

4. Determine the solutions to the equation

for 0 ≤ 𝑥 ≤ 4𝜋 accurate to two decimal places. ✓✓✓✓

5. If the functions y = sin x and y = cos x are both subjected to a horizontal compression by transformation would map the resulting sine curve onto the resulting cosine curve? ✓✓

6. Simplify

. ✓✓

7. Solve 32𝑥 = 72𝑥−1 for x. Round your answer to two decimal places.✓✓✓✓

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1 , what 2

8. If f(x) = - 4 sin x + 6 and g(x) = 3x + 5, what is the maximum value of f (g(x))?.✓✓

Part C – Communication [COMM – 20 marks] **Show ALL of your work and fully LABEL your graphs. 1. Determine an equation in factored form for the polynomial function represented by the graph. ✓✓

2. Explain how you know this is NOT the graph the reciprocal function of f(x) y 5 4 3 2 1

–5 –4 –3 –2 –1 –1

1

2

3

4

5

x

–2 –3 –4 –5

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(x

2) . ✓✓

1

3. Sketch a graph of 𝑦 = −3 cos ( 𝑥 − 2𝜋) + 1 for 2 List below 3 points.

.✓✓✓

𝜋

4. Sketch a graph of 𝑓(𝑥) = 3 sin [2 (𝑥 − 6 )] + 1 for two cycles. ✓✓✓ List below 3 points.

5. Graph the function 𝑓(𝑥) = log(𝑥 + 3) − 1. Identify the domain, the range, and the equation of the vertical asymptote. ✓✓✓✓

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6. Sketch an even function with all of the following features: ✓✓✓✓ • • • •

Absolute maximum at 8 Zeroes at x = 2 and x = –2 y-intercept at 4 3 turning points

7. Let h(x) =

3

2 x − 1 . Define two functions f(x) and g(x), where h(x) = f( g(x)). ✓✓

6.

Part D – Application [APP – 20 marks] 1. Solve each equations for x, where 0 ≤ 𝑥 ≤ 2𝜋. ✓✓& ✓✓ 3 a) cos2 𝑥 − = 0 b) sin2 𝑥 + 3 sin 𝑥 + 2 = 0 4

2. The average cost, in dollars, of producing a toy where x represents the number of toys produced per 100 is given by

.

a) Determine the following, to two decimal places. i) The rate at which the average cost is changing between the production levels of 200 to 300 toys. ✓✓ ii) The rate at which the cost is changing at a production level of 400 toys. ✓✓

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b) What does the sign of your answer to part a), subpart ii), indicate? ✓

3. The population of a town is increasing at a rate of 8.2% per year. The city council believes they will have to add another elementary school when the population reaches 100 000. If there are currently 76 000 people living in the town, how long do they have before the new school will be needed? ✓✓

4. The vertical position, h, in metres, of a Ferris wheel is modeled using the function 𝜋

ℎ(𝑡) = 5 cos [5 (𝑡 − 15)] + 10, where t is the time, in seconds. (a) Determine the period and amplitude of this function. ✓✓

(b) Calculate the height at 𝑡 = 2 s. ✓

(c) Determine the first time that the height is 15 m. ✓✓

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(d) When is the next time that this height is reached again? ✓✓

5. Water will be both added to and drained from a bathtub at the same time for 5 minutes. The total amount of water added to the bathtub in litres will be modelled by the function f(t) = 5t, where t is the time elapsed in minutes. The total amount of water drained from the bathtub in litres will be modelled by the function g(t) = t2, where t is the time elapsed in minutes. How many litres of water will be in the bathtub after 4 minutes? ✓✓

Exam Summary Knowledge & Understanding

Thinking & Invest.

/20

Communication

/24

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Application

/20

/20...