Math 30-2
Name: _______________________
Unit 2 – Polynomial & Sinusoidal Functions
REVIEW
1. The possible number of solutions that a cubic function can have is
A. 0
C. 1 or 2
B. 3
D. 1, 2 or 3
2. Determine the following characteristics of the polynomial function
3
2
f (x)  x  4x  5
Sketch the graph to support your solutions.
a) number of x-intercepts: _____
b) y-intercept: ____
c) end behaviour: __________________
d) domain: _______
e) number of turning points: ____
f) range: _________
3. The volume of water in a cylindrical water tank being drained can be modelled by the function:
v(t) = –3.8t + 225 where v represents the volume in litres and t represents the time in minutes.
How long will it take for the volume to get to 0 L? Round your answer to the nearest minute.
4. Describe the characteristics of the trend in the data.
Degree of function:________ Trend in data: ___________
Math 30-2
Name: _______________________
5. Describe the graph y  x(x  3)2 . Use the words, x-intercept(s), y-intercept, minimum, maximum, trend
in the graph, domain and range in your description.
6. A snowboard company uses the relation: P(x)  210x  60x 2 to model its profits. In the model, P is
the profit in thousands of dollars andx is the number of snowboards in thousands.
a. How much profit should the company expect if they manufacture 2000 snowboards?
MULTIPLE CHOICE
b. The maximum profit of snowboards sold can be found at what point of the graph?
A. x-intercept
C. origin
B. y–intercept
D. vertex
7. Which of the following graphs would most likely represent the graph of a cubic function?
Math 30-2
Name: _______________________
8. Identify the correct polynomial function for this graph. Justify your reasoning.
i) y  
2
x 1
3
iv) y 
1
x
2
2
ii) y  x  x  4
v) y  2(x  2)(x 1)
x 3  4x 2
iii) y  
2
vi) y 
a) Graph 1 matches Equation
because
b) Graph 2 matches Equation
because
c) Graph 3 matches Equation
because
1 3
(x  x  1)
3
Math 30-2
Name: _______________________
9. Write an equation for a polynomial function that satisfies each set of characteristics. Sketch the
graph and state the degree of your polynomial. Explain your reasoning.
a) one turning point, a maximum value, y-intercept of 3
b) cubic, positive end direction, y-intercept of -4.
Math 30-2
Name: _______________________
10. Determine the equation of the linear regression function for the data. Round all values to the
nearest hundredth.
11. Determine the equation of the quadratic regression function for the data. Round all values to the nearest
hundredth.
12. Determine the equation of the cubic regression function for the data to the thousandth. Use the
equation to extrapolate the value of y when x = 50. Round your answer to the nearest whole number.
13. Shane tracked the depth of the water at an ocean marina one afternoon. His data is summarized in
the table below.
a) Determine the equation for the cubic regression function that models this data. Let x be the
number of hours since 12:00 and y be the depth in feet.
b) Interpolate the depth of the water at 17:00. Show your work.
c) Extrapolate the depth of the water at 10:30 that morning. Show your work.
Math 30-2
Name: _______________________
14. A research company summarized the average time spent everyday by teenagers watching television.
a) Create a scatter plot, and draw a curve of best fit for the data using quadratic regression.
b) Use your graph to estimate the average amount of time spent watching television in 2008.
Math 30-2
15. Identify the domain of y  2sin x  1 .
16. Determine the period of the following graph.
17. Determine the midline of the following graph.
A. x  1
B. x  4
C. x  5
D. x  6
18. Determine the amplitude of the following graph.
A. -1
B. 4
C. 5
D. 6
19. Determine the range of the following graph.
A. x R
B. 6  x  4
C. y R
D. 6  y  4
Name: _______________________
Math 30-2
Name: _______________________
20. A sinusoidal graph has a maximum at the point (–40, 3) and a midline of y = –12. Determine
the amplitude of the graph.
21. Determine the amplitude of the following function. y  10 cos 4(x  180)  2
22. Determine the midline of the following function. y  5sin1.5(x  60)  5
23. Determine the period of the following function. y  5sin1.5(x  60)  5
24. Determine the range of the following function. y  5sin1.5(x  60)  5
25. A seat’s position on a Ferris wheel can be modelled by the function
y  18cos2.8(x 1.2)  21
where y represents the height in feet and x represents the time in minutes.
a. Determine the diameter of the Ferris wheel. (HINT: The chair will start at the bottom of the
Ferris Wheel and go upto the top)
b. Which parameter(s) of the equation change if the diameter of the Ferris wheel changes to
24 feet. Explain your answer.
c. Determine the time that elapses between the two minimum heights.
Math 30-2
Name: _______________________
26. The following data set is sinusoidal. Determine the missing value from the table.
27. Determine the equation of the sinusoidal function for the data. Round values to the nearest
tenth.
28. The height of a mass attached to a spring can be modelled by the sinusoidal function
h(t)  53.5  4.2 cos 23.5t
where h(t) represents the height in centimetres and t represents the time in seconds.
a. What is the height of the mass, to the nearest tenth of a centimetre, after the first minute?
b. Calculate, to the nearest tenth of a centimeter, the height of the mass at the moment of
release.
29. Brianna’s position on a Ferris wheel can be modelled by the graph below.
where h(t) represents her height in metres and t represents the time in minutes.
a. How much higher is she after 30 s than at the start of the ride? Round your answer to the
nearest tenth of a metre.
b. What is the maximum height of Brianna above the ground?
Math 30-2
Name: _______________________
30. Jeremy’s gymnastics coach graphs one particular series of jumps. Describe
Jeremy’s jumps using the graph. Show your work.
Amplitude: ______________ Midline: _____________
Range: _________________ Period: ______________
31. Sketch a possible graph of a sinusoidal function with the following set of characteristics.
Explain your decision.
Domain: {x | 0 ≤ x ≤ 4, x ∈ R}
Range: {y | 1 ≤ y ≤ 5, y ∈ R}
Period: 8
y-intercept: 1
Math 30-2
Name: _______________________
32. Kira is sitting in an inner tube in the wave pool. The depth of the water below her, in terms of
time, during a series of waves can be represented by the graph shown.
a) How long does it take for one complete wave to pass? Show
your work.
b) What is the approximate depth of the water below Kira
after 4 s?
c) What is the depth of the water below Kira at 9 s? Assume that the waves continue at the
same rate. Explain your answer.
33. Match each graph with the corresponding equation below. Explain your answers.
i) y = 2 cos 2.5(x – 0.8) – 1
iv) y = 3 cos 2.5(x – 0.6) – 1
ii) y = 3 cos 2(x – 0.8) + 1
v) y = 3 cos 2.5(x – 0.6) + 1
iii) y = 3 cos 2(x – 0.8) – 1
vi) y = 2 cos 2(x – 0.6) – 1
Math 30-2
Name: _______________________
34. Describe the graph of the function y 
1
sin(2x)  3.5 by stating the amplitude, equation of
10
its midline, range, and period. Show your work.
Amplitude: _________________ Midline: _____________
Range: ____________________ Period: ______________
35. A competitive gymnast’s coach analyzes one particular series of jumps. These jumps can be
 
t  9.8 , where h(t) represents the height
 2 
modelled by the sinusoidal function h(t)  11.5sin 
of the gymnast in feet and t represents the time in seconds.
a) What is the maximum height of the gymnast jumps? Explain how you know.
b) How often does the gymnast jump? Explain how you know.
36. The following table gives the population of a predator animal over a decade.
Year
Population
1
165
2
158
3
174
4
198
5
212
6
216
7
205
8
184
9
162
a) Determine the sinusoidal equation for the data above in the form y  asin(bx  c)  d ,
where a, b, c, and d have been rounded to the nearest hundredth.
b) Use your graph to estimate the population in the 12th year.
10
161