MHF 4U
Polynomial Functions
Course End Review
Name: ________________
1.
State the degree and the leading coefficient of each polynomial function.
a)
f(x) = 2x3 + 3x – 1
b) g(x) = 5x – 6
c)
h(x) = x(x – 3)( x + 1)( x2 – 2)
d) r(x) = 21 – 2x + 4x2 – 6x3
2.
Sketch a graph of each polynomial function.
a) y = 2x(x + 3)(x – 4)
b) y = –3(x – 2)(x + 4)(x2 – 1)
2
The zeros of a function are –4, – , and 3. Determine an equation for the function if it has
5
y-intercept 8.
3.
4.
Determine algebraically, if each polynomial function is even or odd.
Graph the functions using TI-nspire to verify your answer.
a) f(x) = 3x5 – 2x3 + x
b) g(x) = 2x4 + 3x3 – 2x – 6
5.
i) Describe the transformations that must be applied to the graph of each power function, f(x), to
obtain the transformed function. Then write the corresponding equation.
ii) State the domain and range of each transformed function.
1
a) f(x) = x4, y = –2f(x – 1) + 4
b) f(x) = x3, y = f(2x + 6) – 5
3
Write an equation for the function that is the result of each set of transformations.
1
a) f(x) = x5 is stretched vertically by a factor of 5, compressed horizontally by a factor of
and
4
translated 2 units to the left and 1 unit down.
1
b) f(x) = x6 is compressed vertically by a factor of , reflected in the y-axis, and translated 4
2
units to the right and 3 units up.
6.
7.
The population of a small town, p, is modelled by the function p(t) = 10 050 + 225t – 20t2, where
t is the time in years from 2000. Determine the AROC of the population from year 2009 to 2014.
Polynomial Equations and Inequalities
8.
i) Use the remainder theorem to determine the remainder for each division.
ii) Perform each division. Express the result in quotient form. Identify all restrictions on the
variable.
a) x3 + 4x2 – 3 divided by x – 2
b) 3x3 – 5x2 + 2x – 6 divided by x – 5
9.
Determine the value of k such that when f(x) = 3x5 – 4x3 + kx2 – 1 is divided by x + 2, the
remainder is –5.
10.
For what value of m will the polynomial P(x) = 2x3 + mx2 – 4x + 1 have the same remainder when
it is divided by x + 2 and by x – 3?
11.
Factor each polynomial.
a) x3 + x2 – 10x + 8
b) 3x4 + x3 – 14x2 – 4x + 8
c) x3 – 3x2 – 9x + 27
d) 4x3 + 4x2 – 25x – 25
12.
Determine the value of b such that x + 4 is a factor of 2x3 – 4x2 + bx – 8.
MHF 4U
Name: ________________
13.
Determine the value of k such that 3x – 2 is a factor of x3 + kx2 – 5x + 3.
14.
Determine the x-intercepts of each polynomial function.
a) y = 27x3 – 64
b) g(x) = x4 – 29x2 + 100
15.
Determine the roots of each polynomial equation.
a) (x2 – 3x – 10)(2x2 + 8) = 0
b) (5x2 – 125)(3x3 – 81) = 0
16.
a) Determine an equation for the family of cubic functions with zeros –
1
, 2, and 6.
2
b) Write equations for two functions that belong to the family in part (a).
c) Determine an equation for the member whose graph passes through the point (–1, 42).
17.
a) Determine an equation, in simplified form, for the family of cubic functions with zeros –3,
1 + 6 , and 1 - 6 .
b) Determine an equation for the member of the family whose y-intercept is –10.
Rational Functions
18.
19.
20.
21.
22.
Determine the equations for the vertical and horizontal asymptotes of each function.
1
3
a) f (x) =
b) g(x) = 2x - 5
x +5
Determine an equation to represent the graph of the function given on the right.
Sketch a graph of each function.
4
a) f (x) =
x-2
b) g(x) = -
1
x +1
For each function, state: i) the domain and range and ii) the x- and y-intercepts
5
1
a) f (x) = b) g(x) =
4x - 7
x-5
For each function,
i) determine the equations of the asymptotes
ii) determine the x- and y-intercepts
iii) sketch the graph
iv) state the domain and range
v) list the intervals over which the function is increasing
vi) list the intervals over which the slope of the graph is increasing
1
8
a) f (x) =
b) g(x) = - 2
(x + 2)(x -1)
x -4
c) g(x) =
4
x +4
2
23.
A function that is the reciprocal of a quadratic function has vertical asymptotes x = –2 and x = 6.
It has a horizontal asymptote y = 0 and the function is positive over the interval (–2, 6). Write an
equation for this function.
24.
Summarize the key features of each function. Then, sketch a graph of the function.
x-3
4 - 3x
a) f (x) =
b) g(x) =
x+2
2x +1
MHF 4U
25.
Name: ________________
ax + b
A function of the form f (x) =
has the following features:
cx + d
3
• x-intercept –1 • y-intercept
• vertical asymptote x = –2 • horizontal asymptote y = 3
2
Determine an equation for this function.
26.
Determine an equation of the function whose graph is shown.
27.
Solve algebraically.
6
a) 3 = 2
2x - x - 4
b)
28.
2x - 3 2x + 7
³
x +5
x-3
 4t + 3 
The population of a town can be modelled by the function P (t ) = 20 
 , where P is the
 2t + 5 
population, in thousands, and t is the time, in years, after the year 2000 (t > 0).
a) What is the population in the year 2000?
b) In what year is the population 30 000?
c) Town planners claim that they need not plan for a population above 40 000. Does the model
support this conclusion? Explain.
Trigonometry
29.
Determine the approximate degree measure, to the nearest tenth, for each angle.
a) 3.62
b) 0.54
c) 5.43
d) 1.75
30.
Determine the approximate radian measure, to the nearest hundredth, for each angle.
a) 59°
b) 243°
c) 124°
d) 425°
31.
Determine the exact radian measure of each angle.
a) 125°
b) 80°
c) 16°
32.
Given that tan
7π
= –cot x, determine the measure of angle x, to two decimal places.
9
33.
Given that cos
3π
= sin a, apply a cofunction identity to determine the exact measure of angle a.
7
34.
Prove that
35.
Determine an exact value for each expression.
π
π
sin tan
4π
5π
3π
6
3
a)
b) sec
cot
– tan
π
3
6
4
csc
4
tan x - tan y
= –tan x tan y.
cot x - cot y
MHF 4U
36.
37.
Name: ________________
3
Angle a lies in the second quadrant and angle b lies in the third quadrant such that cos a = 5
24
and tan b =
. Determine an exact value for:
7
a) cos(a +b)
b) sin(a – b)
c) sin 2a
π
12
radians with the wall. Use a compound angle formula to determine an exact expression for the
minimum distance that the foot of the ladder can be placed from the wall so that the ladder is
standing safely.
A 15-m ladder leaning against a wall is in an unsafe position if it makes an angle of less than
Trigonometric Functions
38.
39.
The graph of a cosine function has a maximum of –1 and a minimum of –9.
a) Determine the amplitude of the function.
b) Determine the vertical translation of the function.
Write an equation of a sinusoidal function which has a period of π and a phase shift of +
π
7π
and ends at x =
.
6
4
π
.
4
40.
One cycle of a sine function begins at x = –
41.
Determine all values of x in the interval [0, 2π] such that sec x = –6. Round your answer to two
decimal places.
42.
Determine the amplitude, period, phase shift, and vertical translation, if it exists, for each function
with respect to y = cos x.
é  3p ù
1
a) y = –4 cos[2(x + 2.5)]
b) y = cos ê3 x + ú – 5
4
4 û
ë 
a) Determine the period of the function.
b) What value of k is required in the equation of the function?
p
1
c) y = –cos  x -  + 3.5
3
2
43.
Determine exact solutions for the equation 2 sin 2x = 0 in the interval x  [0, 2π].
44.
A cosine function has a maximum value of 7, a minimum value of 1, a period of
3π
rad to the right.
4
a) Write an equation for the function.
b) Graph the function and verify that it satisfies the properties given.
shift of
45.
π
, and a phase
2
Determine approximate solutions for each equation in the interval x  [0, 2π], to the nearest
hundredth of a radian.
3
i) sin x – 0.8 = 0
ii) tan x – = 0
iii) 4 csc x + 5 = 0
4
MHF 4U
Name: ________________
46.
Determine approximate solutions for each equation in the interval x  [0, 2π], to the nearest
hundredth of a radian.
49
a) sec2x –
=0
b) tan2x – 2.25 = 0
c) 64 sin2x – 25 = 0
9
47.
Determine the solutions for the equation 2 csc2x – 9 cscx + 10 = 0 in the interval x  [0, 2π].
Give exact solutions where possible. If not possible, round to the nearest hundredth of a radian.
Exponential and Logarithmic Functions
48.
1
a) Graph the function f  x  =   . Identify the key features of the graph (domain, range,
2
intercepts, intervals for which the function is positive and intervals for which it is negative,
intervals over which the function is increasing and intervals for which it is decreasing, equation
of the asymptote).
b) Graph f -1 on the same grid as f by reflecting the graph of f in the line y = x.
x
c) Identify the key features of f -1 .
49.
Rewrite each equation in logarithmic form.
a) 54 = 625
b) 4x = 12
c) y = 123
50.
Rewrite each equation in exponential form.
a) x = log8
b) 4 = log 5 x
c) 7 = log b 200
51.
Evaluate without using a calculator.
 1 
a) log 
b) log 4 64

 1000 
c) log 2 0.25
d) 2log1000
52.
a) What is the value of x log x 6 ? Justify your answer.
b) Is your answer to part a) true for all real values of x? Explain.
53.
Determine an equation for the graph shown to the right.
54.
Sketch the graph of y = -4log  x + 5  - 2 by hand using your knowledge
of transformations.
55.
Evaluate WITHOUT using a calculator.
 5
1
a) log 4  64 6 
b) log 5


100


56.
Solve for x. Round answers to two decimal places.
a) x = log 4 25
b) 3x = 12
c) 15 = x 7.6

d) 3000 = 1500 1+ x

8
MHF 4U
57.
Name: ________________
 I 
The magnitude, M, of an earthquake on the Richter scale is given by M = log   , where I/I0
 I0 
represents the intensity of the earthquake.
a) The magnitudes of two earthquakes are 4.7 and 7.1. How many times as intense was the stronger
earthquake than the less severe one?
b) An earthquake is detected that is 450 times as intense as an earthquake with a magnitude of 5.2.
What is the magnitude of the new earthquake?
Exponential and Logarithmic Equations
58.
Write each as a power of 8.
59.
Solve.
a) 512
a) 4
2 x+5
= 32
b)
1
64
b) 27
4-x
c)
2-3x
1
= 
9
2x
Solve. Round answers to two decimal places. 5 p+4 = 2 4-5 p
61.
Solve. Check for inadmissible roots.
a) 32 x - 4(3) x + 3 = 0
b) 2 x = 2 + 3(2)- x
62.
Evaluate using the laws of logarithms.
a) log 4 12 - log 4 3
b) 3log 6 + 2log5- log54
63.
Write log 3 5 + 2log 3 ( x - 2) as a single logarithm.
64.
Simplify and state restrictions necessary on the variable.
a) log(x 2 - 4x -12) - log(3x -18)
b) log(x 3 - 27) - log(x - 3)
65.
Solve. Check for inadmissible roots.
a) log 3 3x + 7 = 2

Combined Functions


32
c) 4 2 x-5 = 7 x
60.

3

b) log 5 2x +1 = 1- log 5 x + 2

66.
Use the graph shown to draw a graph of each.
a) y = f (x) + g(x)
b) y = g(x) - f (x)
67.
If f (x) = 2x + 3, g(x) = x 2 + x , and h(x) = x 2 - 3 , develop
an algebraic model for each of the following.
a) y = f ( x )  g ( x)
b) y = f (x) - g(x) + h(x)
68.
If f ( x ) = 2 x + 3 , g ( x ) = x 2 + x , and h ( x ) = x 2 - 3 , develop algebraic models for each.
a) y = f (h(x))
b) y = (g o f )(x)
c) y = f ( f -1 (x))