Download Final Exam Practice - Berkeley City College

Berkeley City College
Practice Problems
Math 1 Precalculus - Final Exam Preparation
Name__________________________________________
Please print your name as it appears on the class roster.
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
List the intercepts for the graph of the equation.
1) 4x2 + y 2 = 4
1)
Objective: (1.2) Find Intercepts from an Equation
2) y = x 2 + 13x + 40
2)
Objective: (1.2) Find Intercepts from an Equation
Write the standard form of the equation of the circle.
3)
3)
y
(3, 5)
(7, 5)
x
Objective: (1.4) Write the Standard Form of the Equation of a Circle
Write the standard form of the equation of the circle with radius r and center (h, k).
4) r = 6; (h, k) = (2, -3)
4)
Objective: (1.4) Write the Standard Form of the Equation of a Circle
Find the center (h, k) and radius r of the circle with the given equation.
5) 5(x + 6)2 + 5(y + 2)2 = 30
5)
Objective: (1.4) Write the Standard Form of the Equation of a Circle
Find the value for the function.
6) Find f(x - 1) when f(x) = 4x2 - 3x + 3.
6)
Objective: (2.1) Find the Value of a Function
Instructor: K Pernell
1
2
7) Find f(x + 1) when f(x) = x - 8 .
x- 2
7)
Objective: (2.1) Find the Value of a Function
Find the domain of the function.
8) g(x) = 3x
x2 - 25
8)
Objective: (2.1) Find the Domain of a Function Defined by an Equation
9) h(x) =
x-1
x3 - 81x
9)
Objective: (2.1) Find the Domain of a Function Defined by an Equation
10) f(x) =
13 - x
10)
Objective: (2.1) Find the Domain of a Function Defined by an Equation
For the given functions f and g, find the requested function and state its domain.
11) f(x) = 7x - 4; g(x) = 9x - 9
Find f ∙ g.
11)
Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions
Solve the problem.
12) Find (fg)(4) when f(x) = x - 3 and g(x) = -5x2 + 12x - 4.
12)
Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions
13) Find
f
(-2) when f(x) = 2x - 5 and g(x) = 3x2 + 14x + 4.
g
13)
Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions
Find and simplify the difference quotient of f, f(x + h) - f(x) , h≠ 0, for the function.
h
14) f(x) = 3x2
14)
Objective: (2.1) Form the Sum, Difference, Product, and Quotient of Two Functions
2
The graph of a function f is given. Use the graph to answer the question.
15) For what numbers x is f(x) > 0?
15)
100
100
-100
-100
Objective: (2.2) Obtain Information from or about the Graph of a Function
16) How often does the line y = 1 intersect the graph?
16)
5
5
-5
-5
Objective: (2.2) Obtain Information from or about the Graph of a Function
Answer the question about the given function.
17) Given the function f(x) = 5x2 + 10x + 8, is the point (-1, 3) on the graph of f?
17)
Objective: (2.2) Obtain Information from or about the Graph of a Function
18) Given the function f(x) = 4x2 + 8x - 6, is the point (-2, 2) on the graph of f?
18)
Objective: (2.2) Obtain Information from or about the Graph of a Function
Find the average rate of change for the function between the given values.
19) f(x) = x 2 + 7x; from 1 to 5
19)
Objective: (2.3) Find the Average Rate of Change of a Function
Write the equation of a sine function that has the given characteristics.
20) The graph of y = x2, shifted 6 units upward
20)
Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts
21) The graph of y =
x, shifted 7 units to the right
21)
Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts
3
Graph the function by starting with the graph of the basic function and then using the techniques of shifting,
compressing, stretching, and/or reflecting.
22) f(x) = (x - 3)2 - 4
22)
y
10
5
-10
-5
5
10 x
-5
-10
Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts
23) f(x) = (x + 6)3 + 7
23)
y
10
5
-10
-5
5
10 x
-5
-10
Objective: (2.5) Graph Functions Using Vertical and Horizontal Shifts
Use the accompanying graph of y = f(x) to sketch the graph of the indicated equation.
24) y = - 2f(x + 5) + 4
y
y
10
y = f(x)
10
5
5
-10
-5
5
10 x
-10
-5
5
-5
-5
-10
-10
Objective: (2.5) Graph Functions Using Compressions and Stretches
4
10 x
24)
Determine the slope and y-intercept of the function.
25) h(x) = -5x - 3
25)
Objective: (3.1) Graph Linear Functions
Use the slope and y-intercept to graph the linear function.
26) g(x) = -2x + 1
26)
y
5
-5
5
x
-5
Objective: (3.1) Graph Linear Functions
Find the vertex and axis of symmetry of the graph of the function.
27) f(x) = -x2 - 6x + 5
27)
Objective: (3.3) Identify the Vertex and Axis of Symmetry of a Quadratic Function
Determine, without graphing, whether the given quadratic function has a maximum value or a minimum value and
then find that value.
28) f(x) = x 2 + 2x - 2
28)
Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function
29) f(x) = -x2 - 2x - 6
29)
Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function
Solve the problem.
30) You have 220 feet of fencing to enclose a rectangular region. Find the dimensions of the
rectangle that maximize the enclosed area.
30)
Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function
31) You have 324 feet of fencing to enclose a rectangular region. What is the maximum area?
31)
Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function
32) The quadratic function f(x) = 0.0038x2 - 0.45x + 36.90 models the median, or average, age,
y, at which U.S. men were first married x years after 1900. In which year was this
average age at a minimum? (Round to the nearest year.) What was the average age at
first marriage for that year? (Round to the nearest tenth.)
Objective: (3.3) Find the Maximum or Minimum Value of a Quadratic Function
5
32)
Use the figure to solve the inequality.
33) f(x) < 0
16
33)
y
12
8
4
(-4, 0)
-16 -12
-8
(3, 0)
-4
4
8
12
16 x
-4
-8
-12
-16
Objective: (3.5) Solve Inequalities Involving a Quadratic Function
Solve the inequality.
34) x2 - 8x ≥ 0
34)
Objective: (3.5) Solve Inequalities Involving a Quadratic Function
35) x2 - 64 ≤ 0
35)
Objective: (3.5) Solve Inequalities Involving a Quadratic Function
Form a polynomial whose zeros and degree are given.
36) Zeros: -3, -2, 2; degree 3
36)
Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity
37) Zeros: -4, -2, -1, 1; degree 4
37)
Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity
For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis
at each x -intercept.
38) f(x) = 2(x - 7)(x + 5)3
38)
Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity
39) f(x) = 2(x2 + 3)(x + 2)2
39)
Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity
40) f(x) =
1
x(x2 - 5)
3
40)
Objective: (4.1) Identify the Real Zeros of a Polynomial Function and Their Multiplicity
Find the x- and y-intercepts of f.
41) f(x) = (x + 1)(x - 6)(x - 1)2
41)
Objective: (4.1) Analyze the Graph of a Polynomial Function
6
42) f(x) = -x2(x + 6)(x2 + 1)
42)
Objective: (4.1) Analyze the Graph of a Polynomial Function
Find the power function that the graph of f resembles for large values of |x|.
43) f(x) = 7x - x3
43)
Objective: (4.1) Analyze the Graph of a Polynomial Function
Find the vertical asymptotes of the rational function.
3x
44) f(x) =
(x - 4)(x - 8)
44)
Objective: (4.2) Find the Vertical Asymptotes of a Rational Function
45) f(x) =
x-4
45)
16x - x 3
Objective: (4.2) Find the Vertical Asymptotes of a Rational Function
Give the equation of the horizontal asymptote, if any, of the function.
2
46) h(x) = 8x - 5x - 2
5x2 - 4x + 8
46)
Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function
3
47) h(x) = 3x - 4x - 7
2x + 2
47)
Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function
48) g(x) = x + 8
x2 - 49
48)
Objective: (4.2) Find the Horizontal or Oblique Asymptotes of a Rational Function
Find the indicated intercept(s) of the graph of the function.
49) y-intercept of f(x) = (5x - 15)(x - 3)
x2 + 9x- 19
49)
Objective: (4.3) Analyze the Graph of a Rational Function
50) x-intercepts of f(x) = (x - 2)(2x + 9)
x2 + 5x - 5
50)
Objective: (4.3) Analyze the Graph of a Rational Function
7
Graph the function.
51) f(x) =
2x
(x - 3)(x - 5)
51)
y
40
20
-8
-4
4
8
x
-20
-40
Objective: (4.3) Analyze the Graph of a Rational Function
2
52) f(x) = x + x - 30
x2 - x - 20
52)
y
10
8
6
4
2
-10 -8
-6
-4
-2
-2
2
4
6
8
10
x
-4
-6
-8
-10
Objective: (4.3) Analyze the Graph of a Rational Function
Solve the inequality.
53) 2x2 + 3x < 20
53)
Objective: (4.4) Solve Polynomial Inequalities
54)
x- 7 < 0
x+9
54)
Objective: (4.4) Solve Rational Inequalities
55) x + 18 < 9
x
55)
Objective: (4.4) Solve Rational Inequalities
8
56)
8x ≥ 4x
7-x
56)
Objective: (4.4) Solve Rational Inequalities
Use the Rational Zeros Theorem to find all the real zeros of the polynomial function. Use the zeros to factor f over the
real numbers.
57) f(x) = x4 - 24x2 - 25
57)
Objective: (4.5) Find the Real Zeros of a Polynomial Function
58) f(x) = x3 + 3x2 - 4x - 12
58)
Objective: (4.5) Find the Real Zeros of a Polynomial Function
59) f(x) = 4x3 - 3x2 + 16x - 12
59)
Objective: (4.5) Find the Real Zeros of a Polynomial Function
Find the intercepts of the function f(x).
60) f(x) = x3 + 2x2 - 5x - 6
60)
Objective: (4.5) Find the Real Zeros of a Polynomial Function
61) f(x) = -x2(x + 6)(x2 + 1)
61)
Objective: (4.5) Find the Real Zeros of a Polynomial Function
Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f.
62) Degree 4; zeros: 4 - 5i, 8i
62)
Objective: (4.6) Use the Conjugate Pairs Theorem
Form a polynomial f(x) with real coefficients having the given degree and zeros.
63) Degree: 3; zeros: -2 and 3 + i.
63)
Objective: (4.6) Find a Polynomial Function with Specified Zeros
64) Degree: 4; zeros: -1, 2, and 1 - 2i.
64)
Objective: (4.6) Find a Polynomial Function with Specified Zeros
For the given functions f and g, find the requested composite function value.
65) f(x) = x + 5, g(x) = 2x;
Find (f ∘ g)(0).
65)
Objective: (5.1) Form a Composite Function
66) f(x) = 2x + 4, g(x) = 2x2 + 1;
Find (g ∘ g)(1).
66)
Objective: (5.1) Form a Composite Function
67) f(x) = 2x + 7, g(x) = -2/x;
Find (g ∘ f)(3).
67)
Objective: (5.1) Form a Composite Function
For the given functions f and g, find the requested composite function.
68) f(x) = 7x + 6, g(x) = 5x - 1;
Find (f ∘ g)(x).
Objective: (5.1) Form a Composite Function
9
68)
69) f(x) =
3 , g(x) = 8 ;
x- 1
3x
Find (f ∘ g)(x).
69)
Objective: (5.1) Form a Composite Function
70) f(x) =
x + 4, g(x) = 8x - 8;
Find (f ∘ g)(x).
70)
Objective: (5.1) Form a Composite Function
Indicate whether the function is one-to-one.
71) {(5, -3), (6, -3), (7, -7), (8, 9)}
71)
Objective: (5.2) Determine Whether a Function Is One-to-One
72) {(4, 5), (-5, -4), (8, -3), (-8, 3)}
72)
Objective: (5.2) Determine Whether a Function Is One-to-One
Decide whether or not the functions are inverses of each other.
73) f(x) = 8x - 5, g(x) = x + 8
5
73)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
74) f(x) = 2x - 2, g(x) = 1 x + 1
2
74)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
75) f(x) = 2x2 + 1, g(x) =
x-1
2
75)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
The function f is one-to-one. Find its inverse.
76) f(x) = x2 + 4, x ≥ 0
76)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
77) f(x) = 5x - 7
3
77)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
78) f(x) =
3
x+7
78)
Objective: (5.2) Find the Inverse of a Function Defined by an Equation
Solve the equation.
79) 27 - 3x = 1
4
79)
Objective: (5.3) Solve Exponential Equations
10
2
80) 2x - 3 = 64
80)
Objective: (5.3) Solve Exponential Equations
Change the exponential expression to an equivalent expression involving a logarithm.
81) 73 = 343
81)
Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements
to Exponential Statements
82) 52 = x
82)
Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements
to Exponential Statements
Change the logarithmic expression to an equivalent expression involving an exponent.
83) log 1 = -3
2 8
83)
Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements
to Exponential Statements
84) ln x = 4
84)
Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements
to Exponential Statements
85) ln 1 = -5
e5
85)
Objective: (5.4) Change Exponential Statements to Logarithmic Statements & Logarithmic Statements
to Exponential Statements
Find the exact value of the logarithmic expression.
86) log4 1
64
86)
Objective: (5.4) Evaluate Logarithmic Expressions
87) log
5
5
87)
Objective: (5.4) Evaluate Logarithmic Expressions
Solve the equation.
88) log3 (x2 - 2x) = 1
88)
Objective: (5.4) Solve Logarithmic Equations
89) 7 + 9 ln x = 4
89)
Objective: (5.4) Solve Logarithmic Equations
90) ln
x +5 = 3
90)
Objective: (5.4) Solve Logarithmic Equations
11
91) e
x +7
=5
91)
Objective: (5.4) Solve Logarithmic Equations
Write as the sum and/or difference of logarithms. Express powers as factors.
x3
92) log
4 y8
92)
Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms
7
93) log
16
3 q 2p
93)
Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms
94) log
4
mn
19
94)
Objective: (5.5) Write a Logarithmic Expression as a Sum or Difference of Logarithms
Express as a single logarithm.
95) 3 loga (2x + 1) - 2 loga (2x - 1) + 2
95)
Objective: (5.5) Write a Logarithmic Expression as a Single Logarithm
Use the Change-of-Base Formula and a calculator to evaluate the logarithm. Round your answer to two decimal
places.
96) log8.7 7.6
96)
Objective: (5.5) Evaluate Logarithms Whose Base Is Neither 10 Nor e
Solve the equation.
97) log (3 + x) - log (x - 5) = log 3
97)
Objective: (5.6) Solve Logarithmic Equations
98) log3 x + log3(x - 24) = 4
98)
Objective: (5.6) Solve Logarithmic Equations
Solve the equation. Express irrational answers in exact form and as a decimal rounded to 3 decimal places.
99) 3 x = 41 - x
99)
Objective: (5.6) Solve Exponential Equations
Convert the angle in degrees to radians. Express the answer as multiple of !.
100) 144°
100)
Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees
101) 87°
101)
Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees
12
Convert the angle in radians to degrees.
102) - 11!
6
102)
Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees
103)
34
!
9
103)
Objective: (6.1) Convert from Degrees to Radians and from Radians to Degrees
Find the exact value. Do not use a calculator.
104) cos 2!
104)
Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
105) tan (19!)
105)
Objective: (6.2) Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
106) cos 16!
3
106)
Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°
107) sec 19!
4
107)
Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°
Find the exact value of the expression. Do not use a calculator.
108) tan 7! + tan 5!
4
4
108)
Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°
109) sin 135° - sin 270°
109)
Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°
110) tan 150° cos 210°
110)
Objective: (6.2) Find the Exact Values for Integer Multiples of !/6 = 30°, !/4 = 45°, and !/3 = 60°
Name the quadrant in which the angle θ lies.
111) cot θ < 0, cos θ > 0
111)
Objective: (6.3) Determine the Signs of the Trigonometric Functions in a Given Quadrant
In the problem, sin θ and cos θ are given. Find the exact value of the indicated trigonometric function.
112) sin θ =
1
, cos θ =
4
15
4
Find cot θ.
112)
Objective: (6.3) Find the Values of the Trigonometric Functions Using Fundamental Identities
13
Find the exact value of the indicated trigonometric function of θ.
113) tan θ = - 8 , θ in quadrant II
Find cos θ.
5
113)
Objective: (6.3) Find Exact Values of the Trig Functions of an Angle Given One of the Functions and
the Quadrant of the Angle
Without graphing the function, determine its amplitude or period as requested.
114) y = -2 sin 1 x
Find the amplitude.
3
114)
Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions
115) y = -3 cos 1 x
4
Find the period.
115)
Objective: (6.4) Determine the Amplitude and Period of Sinusoidal Functions
Match the given function to its graph.
116) 1) y = sin 2x
2) y = 2 cos x
3) y = 2 sin x
4) y = cos 2x
A
3
-2π
116)
B
y
3
2
2
1
1
-π
π
2π
x
-2π
-π
-1
-1
-2
-2
-3
-3
C
y
π
2π
π
2π
x
D
y
-2π
3
3
2
2
1
1
-π
π
2π
x
-2π
-π
-1
-1
-2
-2
-3
-3
Objective: (6.4) Graph Sinusoidal Functions Using Key Points
14
y
x
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Answer the question.
117) Which one of the equations below matches the graph?
A) y = 4 cos 2x
B) y = 2 cos 1 x
4
C) y = 4 sin 1 x
2
117)
D) y = 4 cos 1 x
2
Objective: (6.4) Graph Sinusoidal Functions Using Key Points
118) Which one of the equations below matches the graph?
A) y = 2 cos 3x
B) y = 2 sin 1 x
3
C) y = -2 sin 1 x
3
D) y = 2 cos 1 x
3
118)
Objective: (6.4) Graph Sinusoidal Functions Using Key Points
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
119) What is the y-intercept of y = csc x?
119)
Objective: (6.5) Graph Functions of the Form y = A tan(ωx) + B and y = A cot(ωx) + B
Find the exact value of the expression.
120) cos-1
3
120)
2
Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function
15
121) cos-1 -
3
121)
2
Objective: (7.1) Find the Exact Value of an Inverse Sine, Cosine, or Tangent Function
Find the inverse function f-1 of the function f.
122) f(x) = 3 cos x + 6
122)
Objective: (7.1) Find the Inverse Function of a Trigonometric Function
123) f(x) = 2 tan(10x - 3)
123)
Objective: (7.1) Find the Inverse Function of a Trigonometric Function
Find the exact solution of the equation.
124) 4 cos-1 x = !
124)
Objective: (7.1) Solve Equations Involving Inverse Trigonometric Functions
Find the exact value of the expression.
1
125) cos sin-1
2
125)
Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent
Functions
4
126) sec sin-1 9
126)
Objective: (7.2) Find the Exact Value of Expressions Involving the Inverse Sine, Cosine, and Tangent
Functions
Solve the equation on the interval 0 ≤ θ < 2".
127) 2 cos θ + 3 = 2
127)
Objective: (7.3) Solve Equations Involving a Single Trigonometric Function
Solve the equation. Give a general formula for all the solutions.
128) sin θ = 1
128)
Objective: (7.3) Solve Equations Involving a Single Trigonometric Function
Solve the equation on the interval 0 ≤ θ < 2".
129) cos2 θ + 2 cos θ + 1 = 0
129)
Objective: (7.3) Solve Trigonometric Equations Quadratic in Form
130) 2 sin 2 θ = sin θ
130)
Objective: (7.3) Solve Trigonometric Equations Quadratic in Form
Simplify the trigonometric expression by following the indicated direction.
131) Rewrite in terms of sine and cosine: tan x ∙ cot x
Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions
16
131)
132) Multiply
sin θ by 1 + cos θ
1 - cos θ
1 + cos θ
132)
Objective: (7.4) Use Algebra to Simplify Trigonometric Expressions
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
2!
133) 7,
3
133)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
3!
134) -5,
4
134)
Objective: (9.1) Convert from Polar Coordinates to Rectangular Coordinates
The rectangular coordinates of a point are given. Find polar coordinates for the point.
135) (0, -8)
135)
Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates
136) (-
3, -1)
136)
Objective: (9.1) Convert from Rectangular Coordinates to Polar Coordinates
The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ).
137) x2 + 4y 2 = 4
137)
Objective: (9.1) Transform Equations between Polar and Rectangular Forms
138) y 2 = 16x
138)
Objective: (9.1) Transform Equations between Polar and Rectangular Forms
Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation.
139) r = 5
139)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 r
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
17
140) r = 2 sin θ
140)
6
5
4
3
2
1
1 2 3 4 5 6 r
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
141) r sin θ = 5
141)
6
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5
r
Objective: (9.2) Identify and Graph Polar Equations by Converting to Rectangular Equations
Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary.
142) 3 + i
142)
Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form
143) 2 + 2i
143)
Objective: (9.3) Convert a Complex Number between Rectangular Form and Polar Form
Find zw or z as specified. Leave your answer in polar form.
w
!
!
144) z = 8 cos + i sin
6
6
144)
!
!
w = 3 cos + i sin
2
2
Find zw.
Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form
18
145) z = 10(cos 45° + i sin 45°)
w = 5(cos 15° + i sin 15°)
Find z .
w
145)
Objective: (9.3) Find Products and Quotients of Complex Numbers in Polar Form
Write the expression in the standard form a + bi.
146) 2(cos 15° + i sin 15°) 3
146)
Objective: (9.3) Use De Moivre's Theorem
147) (-
3 + i)6
147)
Objective: (9.3) Use De Moivre's Theorem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the equation to the graph.
148) (y + 2)2 = 8(x + 1)
148)
A)
B)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
C)
5
10 x
5
10 x
D)
y
-10
y
10
10
5
5
-5
5
10 x
-10
-5
-5
-5
-10
-10
Objective: (10.2) Analyze Parabolas with Vertex at (h, k)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find an equation for the parabola described.
149) Vertex at (8, 7); focus at (8, 3)
149)
Objective: (10.2) Analyze Parabolas with Vertex at (h, k)
19
Write an equation for the graph.
150)
150)
y
5
-5
5
x
(2, -1)
-5
Objective: (10.3) Analyze Ellipses with Center at (h, k)
Find the center, foci, and vertices of the ellipse.
151) 3x2 + 4y 2 - 36x + 32y + 160 = 0
151)
Objective: (10.3) Analyze Ellipses with Center at (h, k)
Graph the equation.
(x + 1)2 + (y - 2)2 = 1
152)
9
4
152)
y
5
-10
-5
5
10 x
-5
Objective: (10.3) Analyze Ellipses with Center at (h, k)
Find an equation for the hyperbola described.
153) center at (2, 4); focus at (0, 4); vertex at (1, 4)
153)
Objective: (10.4) Analyze Hyperbolas with Center at (h, k)
Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
154) x2 - 25y 2 + 6x + 50y - 41 = 0
Objective: (10.4) Analyze Hyperbolas with Center at (h, k)
20
154)
Graph the hyperbola.
(y + 2)2 - (x - 2)2 = 1
155)
4
9
155)
y
5
-5
5
x
-5
Objective: (10.4) Analyze Hyperbolas with Center at (h, k)
Solve the system of equations by substitution.
156)
5x - 2y = -1
x + 4y = 35
156)
Objective: (11.1) Solve Systems of Equations by Substitution
Solve the system of equations.
157)
x - y + 3z = -16
4x
+ z = -5
x + 2y + z = -3
157)
Objective: (11.1) Solve Systems of Three Equations Containing Three Variables
Perform the row operation(s) on the given augmented matrix.
158) R 2 = -4r1 + r2
1 3 10
4 2 -8
158)
Objective: (11.2) Perform Row Operations on a Matrix
159) (a) R 2 = -4r1 + r2
(b) R 3 = -2r1 + r3
(c) R 3 = 6r2 + r3
1 -3 -5 -2
4 -5 -4 5
2 5 4 6
159)
Objective: (11.2) Perform Row Operations on a Matrix
Write the partial fraction decomposition of the rational expression.
x
160)
(x - 4)(x - 5)
Objective: (11.5) Decompose P/Q, Where Q Has Only Nonrepeated Linear Factors
21
160)
161)
x +3
3
x - 2x2 + x
161)
Objective: (11.5) Decompose P/Q, Where Q Has Repeated Linear Factors
162)
10x + 2
(x - 1)(x2 + x + 1)
162)
Objective: (11.5) Decompose P/Q, Where Q Has a Nonrepeated Irreducible Quadratic Factor
Graph the equations of the system. Then solve the system to find the points of intersection.
163)
y = x2 - 8x + 16
163)
y = -x + 6
y
10
5
-10
-5
5
10 x
-5
-10
Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution
Solve the system of equations using substitution.
164)
x2 + y 2 = 25
164)
x + y = -7
Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution
165)
165)
ln x = 3ln y
3x = 27y
Objective: (11.6) Solve a System of Nonlinear Equations Using Substitution
Solve using elimination.
166)
x2 + y 2 = 145
166)
x2 - y 2 = 17
Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination
22
167)
167)
2x 2 + y 2 = 17
3x2 - 2y 2 = -6
Objective: (11.6) Solve a System of Nonlinear Equations Using Elimination
23
Answer Key
Testname: MATH1_FINAL_EXAM_PRACTICE
1) (-1, 0), (0, -2), (0, 2), (1, 0)
2) (-5, 0), (-8, 0), (0, 40)
3) (x - 5)2 + (y - 5)2 = 4
23)
y
10
4) (x - 2)2 + (y + 3)2 = 36
5) (h, k) = (-6, -2); r = 6
6) 4x2 - 11x + 10
5
-10
x2 + 2x - 7
7)
x- 1
10 x
5
10 x
-10
24)
y
10
5
14) 3(2x+h)
15) [-100, -60), (70, 100)
16) three times
17) Yes
18) No
19) 13
20) y = x2 + 6
-10
-5
-5
-10
x-7
25) m = -5; b = - 3
26)
y
y
10
5
5
-10
5
-5
8) {x|x ≠ -5, 5}
9) {x|x ≠ -9, 0, 9}
10) {x|x ≤ 13}
11) (f ∙ g)(x) = 63x2 - 99x + 36; all real numbers
12) -36
3
13)
4
21) y =
22)
-5
-5
5
10 x
-5
5
-5
-5
-10
27) (-3, 14) ; x = -3
28) minimum; - 3
29) maximum; - 5
30) 55 ft by 55 ft
31) 6561 square feet
32) 1959, 23.6 years old
33) {x|-4 < x < 3}; (-4, 3)
34) (-∞, 0] or [8, ∞)
35) [-8, 8]
36) f(x) = x3 + 3x2 - 4x - 12 for a = 1
24
x
Answer Key
Testname: MATH1_FINAL_EXAM_PRACTICE
37) x4 + 6x3 + 7x2 - 6x - 8
38) 7, multiplicity 1, crosses x-axis; -5, multiplicity 3,
crosses x-axis
39) -2, multiplicity 2, touches x-axis
40) 0, multiplicity 1, crosses x-axis;
5, multiplicity 1, crosses x-axis;
- 5, multiplicity 1, crosses x-axis
41) x-intercepts: -1, 1, 6; y-intercept: -6
42) x-intercepts: -6, 0; y-intercept: 0
43) y = -x3
53) -4 ,
54) (-9, 7)
55) (-∞, 0) or (3, 6)
56) (-∞, 0] or [5, 7)
57) -5, 5; f(x) = (x - 5)(x + 5)(x2 + 1)
58) -3, -2, 2; f(x) = (x + 3)(x + 2)(x - 2)
3
59) ; f(x) = (4x - 3)(x2 + 4)
4
60) x-intercepts: -3, -1, 2; y-intercept: -6
61) x-intercepts: -6, 0; y-intercept: 0
62) 4 + 5i, -8i
63) f(x) = x3 - 4x2 - 2x + 20
44) x = 4, x = 8
45) x = 0, x = -4
8
46) y =
5
64) f(x) = x4 - 3x3 + 5x2 - x - 10
65) 5
66) 19
2
67) 13
47) no horizontal asymptotes
48) y = 0
45
49) 0, 19
9
50) (2, 0), - , 0
2
68) 35x - 1
9x
69)
8 - 3x
51)
y
70) 2 2x - 1
71) No
72) Yes
73) No
74) Yes
75) Yes; Exclude the interval (-∞, 1)
76) f-1(x) = x - 4, x ≥ 4
40
20
-8
-4
4
8
x
-20
77) f-1(x) = 3x + 7
5
-40
52)
12
78) f-1(x) = x3 - 7
79) {3}
80) {3, -3}
81) log 343 = 3
7
82) log x = 2
5
-3
83) 2 = 1
8
y
10
8
6
4
2
-10 -8
-6
-4
5
2
-2
-2
2
4
6
8
10
84) e4 = x
x
85) e-5 = 1
e5
-4
-6
86) -3
1
87)
2
-8
-10
-12
88) {3, -1}
25
Answer Key
Testname: MATH1_FINAL_EXAM_PRACTICE
89) {e
-1/3
}
121)
90) {e6 - 5}
91) {ln 5 - 7}
92) 3 log x - 8 log y
4
4
1 log 16 - 2 log q - log p
93)
3
3
3
7
94)
5!
6
122) f-1(x) = cos-1
x
123) f-1(x) = 1 tan-1
+3
2
10
1 log m + 1 log n - 1 log 19
4
4
4
2
2
2
2
124)
2
3
95) loga a (2x + 1)
(2x - 1)2
125)
96) 0.94
97) {9}
98) {27}
2
3
2
126) 9
127)
ln 4
≈ 0.558
99)
ln 3 + ln 4
65
65
2! , 4!
3
3
128) θ|θ = ! + 2k!
2
4!
100)
5
129) {!}
29!
101)
60
130) 0, !, ! , 5!
6 6
102) -330°
103) 680°
104) 1
105) 0
106) - 1
2
131) 1
1 + cos θ
132)
sin θ
107) 108) 0
134)
109)
7 7 3
133) - ,
2
2
2
2 +2
2
110) - 5
5
2
2 , -5
2
2
!
135) 8, 2
5!
136) 2, 6
3
6
137) r2(cos2 θ + 4 sin2 θ) = 4
138) r sin 2 θ = 16 cos θ
111) IV
112) 15
113) - 5
x-6
3
89
89
114) 2
115) 8!
116) 1B, 2D, 3C, 4A
117) A
118) B
119) none
!
120)
6
26
Answer Key
Testname: MATH1_FINAL_EXAM_PRACTICE
139)
145) 2(cos 30° + i sin 30°)
146) 4 2 + 4 2i
147) -64
148) D
149) (x - 8)2 = -16(y - 7)
6
5
4
3
2
1
-6 -5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
1 2 3 4 5 6 r
150)
(x - 2)2 + (y + 1)2 = 1
16
9
151)
(x - 6)2 + (y + 4)2 = 1
4
3
center: (6, -4); foci: (7, -4), (5, -4); vertices: (8, -4), (4,
-4)
x2 + y 2 = 25; circle, radius 5,
center at pole
152)
y
140)
5
6
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-6
-10
5
10 x
-5
1 2 3 4 5
r
2
153) (x - 2)2 - (y - 4) = 1
3
154) center at (-3, 1)
transverse axis is parallel to x-axis
vertices at (-8, 1) and (2, 1)
foci at (-3 - 26, 1) and (-3 + 26, 1)
1
1
asymptotes of y - 1 = - (x + 3) and y - 1 = (x + 3)
5
5
x2 + (y - 1)2 = 1; circle, radius 1,
center at (0, 1) in rectangular coordinates
141)
5
4
3
2
1
-5 -4 -3 -2 -1
-1
-2
-3
-4
-5
-5
155)
y
1 2 3 4 5
r
5
-5
5
-5
y = 5; horizontal line 5 units
above the pole
142) 2(cos 30° + i sin 30°)
143) 2 2(cos 45° + i sin 45°)
2! + i sin 2!
144) 24 cos
3
3
156) x = 3, y = 8; (3, 8)
157) x = 0, y = 1, z = -5; (0, 1, -5)
3 10
158) 1
0 -10 -48
27
x
Answer Key
Testname: MATH1_FINAL_EXAM_PRACTICE
1 -3 -5 -2
159) 0 7 16 13
0 53 110 88
-4 + 5
160)
x-4 x-5
161)
3 + -3 +
4
x x - 1 (x - 1)2
162)
4 + -4x + 2
x - 1 x2 + x + 1
163)
y
10
5
-10
-5
5
10 x
-5
-10
(2, 4), (5, 1)
164) x = -3, y = -4; x = -4, y = -3
or (-3, -4), (-4, -3)
165) x = 3 3, y = 3 or (3 3, 3)
166) x = 9, y = 8; x = -9, y = 8; x = 9, y = -8; x = -9, y = -8
or (9, 8), (-9, 8), (9, -8), (-9, -8)
167) x = 2, y = 3; x = 2, y = -3; x = -2, y = 3; x = -2, y = -3
or (2, 3), (2, -3), (-2, 3), (-2, -3)
28