Algebra 2 Practice Problems with Answers
The following sections provide many Algebra 2 practice problems along with their solutions to help you master the key concepts of the course.
Linear Equations and Inequalities
1. Solve for x: 3(2x - 5) + 4 = 19
Answer: x = 5
Explanation: Distribute, combine like terms, and solve for x.
2. Solve the inequality: 2(3x + 1) ≤ 5x - 7
Answer: x = -1
Explanation: Distribute, combine like terms, and solve the inequality.
3. Solve the system of equations: 2x + 3y = 11 x - y = 1
Answer: x = 3, y = 2
Explanation: Use the substitution method to solve for x and y.
4. Solve the absolute value inequality: |2x - 3| > 7
Answer: x = < -2 or x = > 5
Explanation: Isolate the absolute value term and solve two separate inequalities.
5. Solve the equation: √(3x - 2) = 5
Answer: x = 29/3
Explanation: Square both sides and solve for x.
Functions
1. Given f(x) = 2x² - 5x + 3, find f(-2).
Answer: f(-2) = 19
Explanation: Substitute -2 for x and simplify.
2. If f(x) = 3x - 1 and g(x) = x² + 2, find (f ∘ g)(x).
Answer: (f ∘ g)(x) = 3(x² + 2) - 1 = 3x² + 5
Explanation: Substitute g(x) for x in f(x) and simplify.
3. Determine the domain of the function: f(x) = √(x - 3)
Answer: Domain: x ≥ 3
Explanation: The radicand must be non-negative.
4. Find the inverse of the function: f(x) = (2x + 1) / 3
Answer: f⁻¹(x) = (3x - 1) / 2
Explanation: Swap x and y, then solve for y.
5. Graph the function: f(x) = |x - 2| + 1
Answer: V-shaped graph with vertex at (2, 1)
Explanation: The graph is a V-shape with the vertex where the expression inside the absolute value equals zero.
Relations
1. Determine if the relation is a function: {(1, 2), (3, 4), (1, 5)}
Answer: Not a function, as 1 is paired with both 2 and 5.
Explanation: In a function, each x-value is paired with at most one y-value.
2. Find the domain and range of the relation: {(0, 1), (2, 3), (4, 5)}
Answer: Domain: {0, 2, 4}, Range: {1, 3, 5}
Explanation: The domain is the set of first coordinates, and the range is the set of second coordinates.
3. Determine if the relation is reflexive, symmetric, or transitive: {(1, 1), (2, 2), (1, 2), (2, 1)}
Answer: Reflexive and symmetric, but not transitive.
Explanation: Check the definitions of reflexive, symmetric, and transitive relations.
4. Compose the relations: R = {(1, 2), (2, 3)} and S = {(2, 4), (3, 5)}
Answer: R ∘ S = {(1, 4), (2, 5)}
Explanation: Find pairs (a, c) such that (a, b) is in R and (b, c) is in S for some b.
5. Find the inverse of the relation: {(1, 3), (2, 4), (5, 6)}
Answer: Inverse: {(3, 1), (4, 2), (6, 5)}
Explanation: Swap the first and second coordinates of each ordered pair.
Cartesian and Coordinate System
1. Plot the points on a coordinate plane: A(2, 3), B(-1, 4), C(0, -2)
Answer: Graph with points A, B, and C plotted.
Explanation: Find the x-coordinate on the horizontal axis and the y-coordinate on the vertical axis for each point.
2. Find the distance between the points (3, 1) and (-2, 5).
Answer: Distance = √((-2 - 3)² + (5 - 1)²) = √(41)
Explanation: Use the distance formula.
3. Determine the midpoint of the line segment joining (1, 2) and (5, 8).
Answer: Midpoint: (3, 5)
Explanation: Use the midpoint formula.
4. Find the slope of the line passing through the points (-1, 3) and (2, -4).
Answer: Slope = (-4 - 3) / (2 - (-1)) = -7/3
Explanation: Use the slope formula.
5. Write the equation of the line with slope 2 and y-intercept -3.
Answer: Equation: y = 2x - 3
Explanation: Use the slope-intercept form.
Sequence
1. Find the 10th term of the arithmetic sequence: 3, 7, 11, 15, ...
Answer: a₁₀ = 39
Explanation: Use the formula for the nth term of an arithmetic sequence.
2. Determine the sum of the first 20 terms of the geometric sequence: 2, 6, 18, 54, ...
Answer: S₂₀ = 2(3²⁰ - 1) / (3 - 1) = 3,486,784,400
Explanation: Use the formula for the sum of the first n terms of a geometric sequence.
3. Find the recursive formula for the sequence: 1, 4, 9, 16, 25, ...
Answer: a₁ = 1, aₙ = aₙ₋₁ + (2n - 1) for n ≥ 2
Explanation: Each term is defined in terms of the preceding term.
4. Determine the explicit formula for the sequence: 2, 5, 8, 11, 14, ...
Answer: aₙ = 3n - 1 for n ≥ 1
Explanation: Each term is defined independently using the term's position.
5. Find the 8th term of the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ...
Answer: F₈ = 21
Explanation: Use the recursive formula to calculate each term successively.
Vector
1. Find the magnitude of the vector v = <3, -4>.
Answer: |v| = √(3² + (-4)²) = 5
Explanation: Use the formula for the magnitude of a vector.
2. Add the vectors u = <2, 1> and v = <-1, 3>.
Answer: u + v = <1, 4>
Explanation: Add the corresponding components.
3. Subtract the vector v = <4, -2> from u = <1, 5>.
Answer: u - v = <-3, 7>
Explanation: Subtract the corresponding components.
4. Find the scalar product of the vectors a = <2, -3> and b = <1, 4>.
Answer: a · b = 2(1) + (-3)(4) = -10
Explanation: Use the formula for the scalar product.
5. Determine the angle between the vectors p = <1, 1> and q = <-1, 1>.
Answer: cos θ = (p · q) / (|p| |q|) = 0, so θ = 90°
Explanation: Use the formula for the angle between two vectors.
Polynomials
1. Find the degree of the polynomial: 3x⁴ - 2x³ + 5x - 1
Answer: Degree: 4
Explanation: The degree is the highest power of the variable.
2. Add the polynomials: (2x² - 3x + 1) + (x² + 4x - 2)
Answer: 3x² + x - 1
Explanation: Add the coefficients of like terms.
3. Multiply the polynomials: (x - 2)(x + 3)
Answer: x² + x - 6
Explanation: Use the distributive property and combine like terms.
4. Divide the polynomials: (2x³ - 5x² + 3x - 1) ÷ (x - 1)
Answer: Quotient: 2x² - 3x + 3, Remainder: 2
Explanation: Use long division or synthetic division.
5. Find the zeros of the polynomial: x³ - 4x² - 7x + 10
Answer: Zeros: x = -1, x = 2, x = 5
Explanation: Factor the polynomial and set each factor equal to zero.
Factoring
1. Factor the expression: 6x² - 7x - 3
Answer: (3x + 1)(2x - 3)
Explanation: Find two numbers whose product is ac and whose sum is b.
2. Factor the difference of squares: 25x² - 16
Answer: (5x + 4)(5x - 4)
Explanation: Use the difference of squares formula.
3. Factor the perfect square trinomial: x² + 6x + 9
Answer: (x + 3)²
Explanation: Use the perfect square trinomial formula.
4. Factor the sum of cubes: 8x³ + 27
Answer: (2x + 3)(4x² - 6x + 9)
Explanation: Use the sum of cubes formula.
5. Factor the expression: 3x⁴ - 48
Answer: 3(x² + 4)(x² - 4)
Explanation: Factor out the GCF and then factor the difference of squares.
Exponents
1. Simplify the expression: (2x³)⁴
Answer: 16x¹²
Explanation: When raising a power to a power, multiply the exponents.
2. Simplify the expression: (3x²y⁻³)³ ÷ (9xy⁻²)²
Answer: x³y⁻⁵
Explanation: Simplify the numerator and denominator separately, then divide.
3. Solve the equation: 4ˣ⁺¹ = 64
Answer: x = 2
Explanation: Set the exponents equal to each other and solve for x.
4. Simplify the expression: (27a⁶b⁻⁹)⅓ ÷ (9a²b⁻³)½
Answer: b⁻¹
Explanation: Simplify the numerator and denominator separately, then divide.
5. Solve the equation: 5 × 2ˣ⁻¹ = 80
Answer: x = 5
Explanation: Isolate the exponential term, then apply the logarithm to both sides.
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