College Algebra Study Guide Questions
Course Introduction
College Algebra is a foundational mathematics course that focuses on the study of algebraic concepts and techniques essential for success in higher-level math and science courses. The course covers topics such as linear and quadratic equations and inequalities, functions and their graphs, polynomial and rational functions, exponential and logarithmic expressions, systems of equations, and complex numbers. Students will develop problem-solving skills, learn to apply algebraic methods to real-world scenarios, and gain a deeper understanding of mathematical relationships necessary for advanced coursework in mathematics, engineering, business, and the sciences.
Recommended Textbook
Precalculus 2nd Edition by John W. Coburn
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1334 Verified Questions
1334 Flashcards
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Chapter 1: Equations and Inequalities
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107 Verified Questions
107 Flashcards
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Sample Questions
Q1) Solve using the most efficient method. x<sup>2</sup> - 9x = -14
A) x = -7, x = -2
B) x = 7, x = -2
C) x = -7, x = 2
D) x = 7, x = 2
Answer: D
Q2) The length of a garden is 5 ft less than twice its width. The area of the garden is 88 ft<sup>2</sup>. Find the length and width of the garden.
Answer: 11 ft, 8 ft
Q3) Solve the absolute value equation. Write the solution in set notation. -2|x - 9| + 10 = 2
A) {3, 15}
B) {5, 13}
C) {0, 18}
D) {-2, 20}
Answer: B
Q4) Solve by completing the square. Write your answer in exact form. 4x<sup>2</sup> + 2x - 3 = 0
Answer: 11ea7f29_4fa7_9292_9ecd_81c71ec5d742_TB3307_11
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Page 3
Chapter 2: Relations, Functions and Graphs
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Sample Questions
Q1) Determine the domain of the sum of f and g.
A) x \(\in\)(-5, 2)
B) x \(\in\) (-?, -5) \(\cup\) (2, ?)
C) x \(\in\) (-?, -5) \(\cup\) (-5, 2) \(\cup\) (2, ?)
D) x\(\in\) (-?, ?)
Answer: C
Q2) Sketch the graph using transformations of a parent function (without a table of values).
f(x) = (x + 3)<sup>2</sup>
Answer: 11ea7f29_4fb8_33f1_9ecd_6934971485ee_TB3307_00 (Gridlines are spaced one unit apart.)
Q3) Determine the value of f(-6) if f(x) = -x<sup>2</sup> - 4x.
A) 30
B) -12
C) 36
D) -40
Answer: B
Q4) Evaluate (pq)(8).
Answer: 11ea7f29_4fc2_4665_9ecd_6192f8e5d277_TB3307_00
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Chapter 3: Polynomial and Rational Functions
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Sample Questions
Q1) Find the vertex.
f(x) = x<sup>2</sup> + 10x - 8
Answer: (-5, - 33)
Q2) State the end behavior of the function. f(x) = -3x<sup>5</sup> + 5x<sup>4</sup> + x<sup>2 </sup>- 2x + 2
A) down/up
B) up/down
C) down/down
D) up/up
Answer: B
Q3) Use synthetic division and the remainder theorem to evaluate P(4) if P(x) = x<sup>3</sup> - 5x<sup>2</sup> - 4x + 20. Verify using a second method.
Answer: 11ea7f29_4fc9_98a8_9ecd_8d2ca986cce7_TB3307_00 Since the remainder is -12, P(4) = -12.
Verification: P(4) = (4)<sup>3</sup> - 5(4)<sup>2</sup> - 4(4) + 20 = -12
Q4) Write the variation equation.
Answer: y = 11ea7f29_4fda_d732_9ecd_17303374ffda_TB3307_11
Q5) Write the variation equation.
Answer: A = 11ea7f29_4fdb_4c66_9ecd_5d253cf92f09_TB3307_11
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Chapter 4: Exponential and Logarithmic Functions
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Sample Questions
Q1) Graph by the following function by translating the basic function y = b<sup>x</sup>, sketching the asymptote, and strategically plotting a few points to round out the graph. Clearly state what shifts are applied. y = 2<sup>x</sup> <sup>+ 1</sup> - 2
Q2) Solve the equation by writing it in exponential form. Answer in decimal form to the thousandths place. ln e<sup>6</sup><sup>x</sup> = -32.4
A) x = 0.027
B) x = 1.686
C) x = -0.644
D) x = -5.400
Q3) Solve the equation using the uniqueness property of logarithms. log(10x + 10) = log 20
Q4) Find the inverse function of the one-to-one function given. {(-1, 1), (-3, 4), (4, 0), (-5, 3), (-2, 2)}
Q5) Write the equation in exponential form. 2 = log<sub> 5 </sub>25
Q6) Determine whether the function is one-to-one. If not, state why. {(3, 4), (1, 7), (8, 3), (-1, 4), (2, 5), (6, -1)}
Q7) t = 8.5
Page 6
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Chapter 5: Introduction to Trigonometric Functions
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Sample Questions
Q1) What is the linear velocity of the wheels in miles per hour?
A) about 64.8 mi/hr
B) about 65.2 mi/hr
C) about 65.6 mi/hr
D) about 66.9 mi/hr
Q2) Name the reference angle \(\theta\)<sub>r</sub> for \(\theta\) = 225°.
A) 15°
B) 25°
C) 35°
D) 45°
Q3) Use the formula for arc length to find the value of the unknown quantity: s = r\(\theta\).
\(\theta\)= 2.1; r = 220 ft.
Q4) Draw the graph of y = 3 sec t by applying observations made in this chapter about the related cosine graph.
Q5) Use a calculator to find the value of tan 34°, rounded to four decimal places.
Q6) Use a reference rectangle and the rule of fourths to draw an accurate sketch through at least one complete cycle where t > 0. y = -2 sin t
Page 7
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Chapter 6: Trigonometric Identities, Inverses, and Equations
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Sample Questions
Q1) Use a cofunction identity to write an equivalent expression. cos 13°
Q2) Write cot x entirely in terms of cos x.
Q3) Rewrite in terms of an expression containing only cosines to the power 1. sin<sup>2</sup> x cos<sup>4</sup> x
Q4) Evaluate using a calculator. Answer to the nearest tenth of a degree. sec<sup>-1</sup> 5.593
Q5) Use algebra and fundamental identities to rewrite the given expression to create a new identity relationship.
8 sec x tan x
Q6) Rewrite as a single expression. sin(7\(\theta\) )cos(3\(\theta\)) + cos(7\(\theta\))sin(3\(\theta\) )
A) cos(10\(\theta\) )
B) cos(4\(\theta\) )
C) sin(10\(\theta\) )
D) sin(4\(\theta\) )
Q7) Verify the equation is an identity using special products and fundamental identities. (1 + cos x)[1 - cos(-x)] = sin<sup>2</sup> x
Page 8
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Chapter 7: Applications of Trigonometry
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86 Flashcards
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Sample Questions
Q1) Solve using the law of sines and a scaled drawing. If two triangles exist, solve both completely.
side a = 23.6 yd
\(\angle\)A = 30°
side c = 47.2 yd
Q2) Compute 2u + 5v.
A) -17i + 7j
B) -16i + 8j
C) -17i + 8j
D) -16i + 7j
Q3) Hal pushes a box full of books 40 ft. If he uses a constant force of 75 pounds, how much work did he do?
Q4) Find the total voltage across the circuit.
A) 74V
B) 75V
C) 76V
D) 77V
Q5) Solve using the nth roots theorem. Approximate answers to four decimal places. x<sup>5</sup> - 1024 = 0
Page 9
Q6) Find a unit vector pointing in the same direction as the vector v = -2i - 5j.
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Chapter 8: Systems of Equations and Inequalities
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Sample Questions
Q1) Solve the linear inequality by shading the appropriate half plane. -3x + y \(\ge\) 0
Q2) Write the system of equations corresponding to the matrix.
Q3) Find any four ordered triples that satisfy the equation.
x - y + z = -3
Q4) Solve the linear inequality by shading the appropriate half plane. 2x + 3y > 6
Q5) The length of a rectangle is 9 in. more than twice its width. If the perimeter of the rectangle is 54 in., find the width of the rectangle.
A) 5 in.
B) 6 in.
C) 7 in.
D) 8 in.
Q6) Determine whether the ordered pair (0, 0) is a solution. 10x - 3y > 5
A) Yes
B) No
Q7) Solve the linear inequality by shading the appropriate half plane. x - y \(\le\) 2
Q8) Solve the linear inequality by shading the appropriate half plane. 3x + y \(\le\) 5
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Chapter 9: Analytical Geometry
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122 Flashcards
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Sample Questions
Q1) Does the arrow hit the target?
A) No
B) Yes
Q2) Classify the equation as that of a circle, ellipse, hyperbola, or none of the above. 4x<sup>2</sup> - 7y<sup>2</sup> = 28
A) Circle
B) Ellipse
C) Hyperbola
D) None of these
Q3) Eliminate the parameter and write the corresponding rectangular form. x = t + 10
Y = t<sup>2</sup> + 5
A) y = x<sup>2</sup> + 105
B) y = x<sup>2</sup> - 95
C) y = x<sup>2</sup> - 20x + 105
D) y = x<sup>2</sup> + 20x + 105
Q4) Find the vertex.
Q5) Show the figure drawn by connecting points A(-3, 1), B(4, -5), C(5, -1), and D(-2, 5) is a parallelogram (opposite sides parallel and equal in length).
Q6) Find the x- and y-intercepts (if they exist).
Page 11
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Chapter 10: Additional Topics in Algebra
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Sample Questions
Q1) Find the number of terms n in the sequence. a<sub>1</sub> = 3, a<sub>n</sub> = 192, and r = 2.
Q2) Use Pascal's triangle and the patterns explored to write the expansion. (5x + 1)<sup>4</sup>
Q3) What is the population after 5 hours?
A) 100
B) 200
C) 640
D) 20,480
Q4) Determine if the sequence given is arithmetic. If yes, name the common difference. 4, 6, 8, 10, 12, . . .
A) arithmetic; d = 2
B) arithmetic; d = -2
C) arithmetic; d = 4
D) not arithmetic
Q5) For S<sub>n</sub> = n(5n - 3), find S<sub>4</sub>, S<sub>5</sub>, S<sub>k</sub>, and S<sub>k</sub><sub> + 1</sub>.
Q6) Use the complementary events to complete the exercise. Three standard dice are rolled. What is the probability the sum is less than 18?
Q7) What is the probability the student answered "No" or is a girl?
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Chapter 12: Review of Basic Concepts and Skills
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112 Flashcards
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Sample Questions
Q1) Factor out the greatest common factor. -18x<sup>4</sup> + 10x<sup>3 </sup>12x<sup>2</sup>
A) -2x<sup>2</sup>(9x<sup>2</sup> + 10x - 12)
B) -2x<sup>2</sup>(9x<sup>2</sup> - 5x + 6)
C) -18x<sup>2</sup>(x<sup>2</sup> + 10x - 12)
D) -18x(x<sup>3</sup> + 10x<sup>2</sup> - 12x)
Q2) Find the binomial square. (4x - 3)<sup>2</sup>
A) 4x<sup>2</sup> - 12x + 9
B) 16x<sup>2</sup> - 12x + 9
C) 16x<sup>2</sup> + 9
D) 16x<sup>2</sup> - 24x + 9
Q3) Identify the number of terms in the expression and the coefficient of each term. -5x<sup>2</sup> + x - 6
Q4) Factor the perfect square trinomial completely. x<sup>2</sup> - 6x + 9
Q5) Factor the trinomial completely. x<sup>2</sup> - 18x + 80
A) (x + 8)(x + 10)
B) (x - 8)(x + 10)
C) (x + 1)(x + 80)
D) (x - 8)(x - 10)
Q6) Factor the trinomial completely. 3x<sup>2</sup> + 20x + 25
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