MHF4U: Test – Functions: Understanding Rates of Change
MHF4U: Unit Tests – Final Review
Name: _________________________________________
Understanding Rates of Change
1. A) What is the average rate of change for the functions [pic] over the interval[pic]? Is the function increasing or decreasing over this interval?
B) Using the Centered-Interval Method, estimate the instantaneous rate of change of the function when[pic]. What does this mean?
2. The path of an arrow shot by an archer is given by the function [pic] where [pic]the height in metres (m) is and [pic] is the time in seconds(s).
A) What is the maximum height reached by the arrow? (Hint: Complete the Square)
B) If the centre of the target is located at a height of 1.1 m. How long after the arrow was released did it take to hit the centre of the target?
3. Estimate the instantaneous rate of change for the function [pic] at [pic] using the difference-quotient method.
4. An automobile enters a road and travels the following distance in metres during the next 10 seconds, where [pic] represents the distance in metres and [pic]time in seconds.
|[pic] |0 |2 |4 |
|[pic] |[pic] | |-2 |
| |[pic] |[pic] |[pic] |
1. Determine the value of [pic] such that when [pic] is divided by [pic] the remainder is -5?
2. Using the remainder theorem, calculate the remainder of [pic]. State any restrictions on the variable as well.
3. Factor fully each of the following polynomials.
A) [pic] B) [pic]
4. Determine all real roots for the following polynomial equations.
A) [pic] B) [pic]
5. A rectangular prism has dimensions 10 cm by 10 cm by 5 cm. When each dimension is increased by the same amount, the new volume of the prism is 1008 [pic]. What are the dimensions of the new prism? Note [pic]
6. Solve each of the following inequalities.
A) [pic] B) [pic][pic]
C) [pic]
7. Using the graph below, identify the approximate intervals where [pic]
8. Sketch the polynomial function [pic] on the graph provided. (For full marks test all max/min values and label axes with an appropriate scale?
Rational Functions
1. For what interval does the reciprocal function of [pic]increase?
2. Determine the equation of a rational function that meets the following conditions:
- Vertical Asymptote at [pic]
- Horizontal Asymptote at [pic]
- Increases on each interval of its domain
- X-intercept is (3,0)
3. The graph below represents the reciprocal function of [pic]. What is the equation of [pic]?
4. Graph the linear function[pic]and its reciprocal function. Then complete the table below for the reciprocal function only.
| |[pic] |
|Domain | |
|Range | |
|Vertical Asymptotes | |
|Horizontal Asymptotes | |
|Invariant Points | |
| | |
|End Behaviours | |
|As [pic] | |
|As [pic] | |
5. Determine all discontinuities for the following rational functions by determining all holes and or vertical and or horizontal and or oblique asymptotes
A) [pic] B)[pic]
6. Sketch the rational function [pic] then determine
Type and equations of all asymptotes
All intercepts (both x and y)
End Behaviours
7. Solve the following rational equations. State all restrictions as well.
A) [pic] B) [pic]
8. Solve the following rational inequality. [pic]
Trigonometric Functions
1. a) Determine the exact values of sec [pic]and tan [pic], if [pic]is a principal angle in quadrant 3 and the sin [pic]= [pic]
b) Determine the value of [pic] in radians rounded to the nearest hundredth.
2. If the central angle within a circle is [pic], what should the radius of the circle be if the arc length is 2.5m?
3. Determine each of the following for the following function. [pic]
Amp: Period: Phase Shift: Vert. Shift:
4. If [pic]is coterminal to[pic]. Determine the principal angle in radian measure. (Note: express as a fraction of [pic])
5. Determine two angles (one positive, one negative) that are coterminal to each of the following. (Note: express in same units as original)
A) [pic] B) [pic]
6. Evaluate each of the following using special triangles. (Note: NO DECIMALS ALLOWED, simplify fully for full marks)
A) [pic] B) [pic]
7. Determine the equivalent expressions ONLY for the trigonometric ratios above in Question 6.
8. Determine the value(s) of [pic] for each of the following. Consider [pic].
A) [pic] B) [pic]
9. Use cofunction identities to write an expression that is equivalent to [pic].
10. Write all equivalent expressions for the following trig ratio, using the related acute angle. (Be sure to include expressions for each of the 4 quadrants)
[pic]
11. Give the measure of the angle [pic], (express in radians as a fraction of [pic])
[pic]
12. A) Sketch and fully label one cycle of both functions[pic] and [pic]on the graph below.
B) State the Domain, Range and period of the transformed function [pic]
13. State the period, amplitude of the following trigonometric functions. Then use this information to write the equation of the function. (Note: Assume no phase shift present)
|X |0 |[pic] |[pic] |[pic] |[pic] |
|Y |-2 |1 |4 |1 |-2 |
14. The sine function can be expressed in the form [pic], describe what you know about [pic]for each of the following statements to be true.
a. The period is greater than [pic].
b. The amplitude is less than 1.
c. The function has no x-intercepts.
Compound, Double Angles and Trig Equations
1. Express the following as a single trigonometric ratio.
A) [pic] B) [pic]
2. Let [pic] and [pic] be third and fourth quadrant angles, respectively, with [pic] and [pic], evaluate each of the following: (NO DECIMALS ALLOWED)
A) [pic]
[
B) [pic]
C) Determine which quadrant the angle [pic]lies in?
[
D) Determine the principal angle [pic]for both [pic] and [pic]in radians. (Round final answers to one decimal place)
3. Determine the exact value of:
A) [pic] B) [pic]
C) [pic] D) [pic]
4. Solve the following trig equations
A) [pic] for [pic]
B) [pic] for [pic]
C) [pic] for [pic]
D) [pic] for [pic]
E) [pic][pic]
5. Given that point P(x,y) lies on the terminal arm of [pic]and that P(x,y) also lies on the “Unit Circle” (radius = 1). (Hint: Include a detailed and label diagram if necessary)
Prove that [pic]
6. Use the appropriate compound angle and double angle formulas to show that[pic].
Logarithmic Functions
1. Graph the exponential function [pic] and its inverse on the graph below. State the equation of the inverse equation in logarithmic form.
2. State the transformations that must be applied to the [pic] to graph [pic]
3. A) Sketch the logarithmic function [pic] on the graph below.
B) State the equation of the Vertical asymptote.
C) State the domain of the function.
D) State the range of the function.
E) With your knowledge of solving logarithmic equations, determine the x-intercept of the function.
4. A) Express the following in exponential form: [pic]
B) Express the following in logarithmic form: (Do not solve): [pic]
5. Express each of the following as a single logarithm.
A) [pic] B) [pic]
[
6. Evaluate.
A) [pic] b) [pic] c) [pic]
d) [pic] e) [pic] f) [pic]
7. Solve each of the following equations.
A) [pic] B) [pic]
C) [pic] D) [pic]
E) [pic] F) [pic]
8. Solve the following system of equations algebraically. (Hint: Substitution Method)
[pic]
ANSWER KEY:
Understanding Rates of Change
1. A) -3 decreasing B) 0 Change direction (Max or Min) 2.A) 4.2m B) t=2.59 sec C) OMIT 3. 14 4. A) 5 m/s B) 5 m/s C) [pic] [pic] 12.25 m/s
Polynomial Equations and Inequalities
1. Remainder = -y 2. Quotient = [pic] Dividend= [pic]
3. [pic] 4. Remainder = -16 [pic] 5. A) [pic] B) [pic]
6. A) [pic] B) [pic] 7. length = 12 Width= 12 Height = 7 cm 8. A) [pic] B) [pic]
C) [pic], [pic] 9. [pic] 10.
Rational Functions
1. [pic] 2. [pic] 3. [pic]
4.
| |[pic] |
|Domain | |
|Range | |
|Vertical Asymptotes | |
|Horizontal Asymptotes | |
|Invariant Points | |
| | |
|End Behaviours | |
|As [pic] | |
|As [pic] | |
5. A) Hole at [pic] B) Hole at [pic], Vert Asy at [pic] Oblique Asy at [pic]
6. 7. A) [pic]restrictions [pic]
B) [pic] restrictions [pic]
8. [pic] and [pic]
Trigonometric Functions
1. A) [pic] , [pic] b) [pic] 2. radius = .63 m
3. amp: [pic] Period: [pic] Phase Shift: [pic] Vert Shift: down 8 4. [pic]
5. A) [pic] B) [pic] 6. A) -2 B) [pic] 7. OMIT 8. A) [pic]
B) [pic] 9. [pic] 10. Q1: [pic], Q3: [pic], Q4: [pic]
11. [pic] 12. [pic] Domain: [pic] Range: [pic] Period: [pic]
13. [pic] 14. A) [pic] B) [pic] C) [pic]
Compound, Double Angles and Trig Identities
1. A) [pic] B) [pic] 2. A) [pic] B) [pic] C) Quad 4 D) [pic], [pic]
3. A) [pic] B) [pic] C) [pic] D) [pic] 4.A) [pic]
B) [pic]C) [pic]D) [pic]
E) [pic]
Logarithmic Functions
1. 2. [pic] Vert. Comp [pic]
Up 3
Hor Comp [pic]
Reflect over y-axis
Right 2
3.B) x=2 4.A) [pic] B) [pic]
C)[pic] 5.A) [pic] B) [pic]
D)[pic] 6. A) [pic] B) 25 C) [pic] D) 4 E) 5 F) -3
E) x= 7.62 7. A) [pic] B) [pic] C) [pic] D) [pic]
E) [pic] F) [pic]
8.[pic]
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[pic]
[pic]
[pic]
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