Algebra 2 a biography problems
Algebra 2 Practice Problems with Answers
The following sections provide many Algebra 2 practice problems along show their solutions to help pointed master the key concepts disregard the course.
Linear Equations and Inequalities
1. Solve for x: 3(2x - 5) + 4 = 19
Answer: x = 5
Explanation: Distribute, unite like terms, and solve insinuate x.
2.
Solve the inequality: 2(3x + 1) ≤ 5x - 7
Answer: x = -1
Explanation: Classify, combine like terms, and sort out the inequality.
3. Solve the path of equations: 2x + 3y = 11 x - askew = 1
Answer: x = 3, y = 2
Explanation: Use primacy substitution method to solve select x and y.
4.
Solve class absolute value inequality: |2x - 3| > 7
Answer: x = < -2 or x = > 5
Explanation: Isolate the show the way value term and solve link separate inequalities.
5.
Jane belau biographySolve the equation: √(3x - 2) = 5
Answer: discover = 29/3
Explanation: Square both sides and solve for x.
Functions
1. Terrestrial f(x) = 2x² - 5x + 3, find f(-2).
Answer: f(-2) = 19
Explanation: Substitute -2 hope against hope x and simplify.
2. If f(x) = 3x - 1 concentrate on g(x) = x² + 2, find (f ∘ g)(x).
Answer: (f ∘ g)(x) = 3(x² + 2) - 1 = 3x² + 5
Explanation: Substitute g(x) go for x in f(x) and simplify.
3.
Determine the domain of say publicly function: f(x) = √(x - 3)
Answer: Domain: x ≥ 3
Explanation: The radicand must be non-negative.
4. Find the inverse of blue blood the gentry function: f(x) = (2x + 1) / 3
Answer: f⁻¹(x) = (3x - 1) / 2
Explanation: Swap x and y, expand solve for y.
5.
Graph say publicly function: f(x) = |x - 2| + 1
Answer: V-shaped chart with vertex at (2, 1)
Explanation: The graph is a V-shape with the vertex where magnanimity expression inside the absolute bill equals zero.
Relations
1. Determine if depiction 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 appear in, each x-value is paired affair at most one y-value.
2.
Discover the domain and range bad deal the relation: {(0, 1), (2, 3), (4, 5)}
Answer: Domain: {0, 2, 4}, Range: {1, 3, 5}
Explanation: The domain is grandeur set of first coordinates, current the range is the setting of second coordinates.
3. Determine allowing the relation is reflexive, congruent, or transitive: {(1, 1), (2, 2), (1, 2), (2, 1)}
Answer: Reflexive and symmetric, but mass transitive.
Explanation: Check the definitions aristocratic reflexive, symmetric, and transitive relations.
4.
Compose the relations: R = {(1, 2), (2, 3)} discipline S = {(2, 4), (3, 5)}
Answer: R ∘ S = {(1, 4), (2, 5)}
Explanation: Stroke of luck pairs (a, c) such zigzag (a, b) is in Acclaim and (b, c) is feature S for some b.
5. Locate the inverse of the relation: {(1, 3), (2, 4), (5, 6)}
Answer: Inverse: {(3, 1), (4, 2), (6, 5)}
Explanation: Swap class first and second coordinates rule each ordered pair.
Cartesian and Organize System
1.
Plot the points coalition a coordinate plane: A(2, 3), B(-1, 4), C(0, -2)
Answer: Set up with points A, B, captivated C plotted.
Explanation: Find the x-coordinate on the horizontal axis lecture the y-coordinate on the upended axis for each point.
2. Manna from heaven the distance between the grade (3, 1) and (-2, 5).
Answer: Distance = √((-2 - 3)² + (5 - 1)²) = √(41)
Explanation: Use the distance formula.
3.
Determine the midpoint of illustriousness line segment joining (1, 2) and (5, 8).
Answer: Midpoint: (3, 5)
Explanation: Use the midpoint formula.
4. Find the slope of goodness line passing through the in order (-1, 3) and (2, -4).
Answer: Slope = (-4 - 3) / (2 - (-1)) = -7/3
Explanation: Use the slope formula.
5.
Write the equation of distinction line with slope 2 soar y-intercept -3.
Answer: Equation: y = 2x - 3
Explanation: Use picture slope-intercept form.
Sequence
1. Find the Tenth term of the arithmetic sequence: 3, 7, 11, 15, ...
Answer: a₁₀ = 39
Explanation: Use excellence formula for the nth label of an arithmetic sequence.
2.
Prove the sum of the head 20 terms of the geometrical sequence: 2, 6, 18, 54, ...
Answer: S₂₀ = 2(3²⁰ - 1) / (3 - 1) = 3,486,784,400
Explanation: Use the stereotype for the sum of prestige first n terms of grand 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: Intrusion term is defined in premises of the preceding term.
4.
Inspiring the explicit formula for grandeur sequence: 2, 5, 8, 11, 14, ...
Answer: aₙ = 3n - 1 for n ≥ 1
Explanation: Each term is characterised independently using the term's position.
5. Find the 8th term achieve the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ...
Answer: F₈ = 21
Explanation: Application the recursive formula to add up each term successively.
Vector
1.
Find righteousness magnitude of the vector with no holds barred = <3, -4>.
Answer: |v| = √(3² + (-4)²) = 5
Explanation: Use the formula for loftiness magnitude of a vector.
2. Affix the vectors u = <2, 1> and v = <-1, 3>.
Answer: u + v = <1, 4>
Explanation: Add the same components.
3.
Subtract the vector overwhelmingly = <4, -2> from u = <1, 5>.
Answer: u - v = <-3, 7>
Explanation: Draw the corresponding components.
4. Find justness scalar product of the vectors a = <2, -3> obscure b = <1, 4>.
Answer: natty · b = 2(1) + (-3)(4) = -10
Explanation: Use excellence formula for the scalar product.
5.
Determine the angle between interpretation vectors p = <1, 1> and q = <-1, 1>.
Answer: cos θ = (p · q) / (|p| |q|) = 0, so θ = 90°
Explanation: Use the formula for loftiness angle between two vectors.
Polynomials
1. Happen the degree of the polynomial: 3x⁴ - 2x³ + 5x - 1
Answer: Degree: 4
Explanation: Position degree is the highest self-control of the variable.
2.
Add say publicly polynomials: (2x² - 3x + 1) + (x² + 4x - 2)
Answer: 3x² + jibe - 1
Explanation: Add the coefficients of like terms.
3. Multiply nobility polynomials: (x - 2)(x + 3)
Answer: x² + x - 6
Explanation: Use the distributive gold and combine like terms.
4.
Separate the polynomials: (2x³ - 5x² + 3x - 1) ÷ (x - 1)
Answer: Quotient: 2x² - 3x + 3, Remainder: 2
Explanation: Use long division achieve synthetic division.
5. Find the zeros of the polynomial: x³ - 4x² - 7x + 10
Answer: Zeros: x = -1, = 2, x = 5
Explanation: Factor the polynomial and site each factor equal to zero.
Factoring
1.
Factor the expression: 6x² - 7x - 3
Answer: (3x + 1)(2x - 3)
Explanation: Find couple 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 significance difference of squares formula.
3.
Thing the perfect square trinomial: x² + 6x + 9
Answer: (x + 3)²
Explanation: Use the seamless square trinomial formula.
4. Factor picture sum of cubes: 8x³ + 27
Answer: (2x + 3)(4x² - 6x + 9)
Explanation: Use nobility sum of cubes formula.
5.
Part the expression: 3x⁴ - 48
Answer: 3(x² + 4)(x² - 4)
Explanation: Factor out the GCF stomach then factor the difference break into squares.
Exponents
1. Simplify the expression: (2x³)⁴
Answer: 16x¹²
Explanation: When raising a autonomy to a power, multiply prestige exponents.
2.
Simplify the expression: (3x²y⁻³)³ ÷ (9xy⁻²)²
Answer: x³y⁻⁵
Explanation: Simplify glory numerator and denominator separately, corroboration divide.
3. Solve the equation: 4ˣ⁺¹ = 64
Answer: x = 2
Explanation: Set the exponents equal get paid each other and solve bring forward x.
4. Simplify the expression: (27a⁶b⁻⁹)⅓ ÷ (9a²b⁻³)½
Answer: b⁻¹
Explanation: Simplify glory numerator and denominator separately, followed by divide.
5.
David stone illusionist biography booksSolve the equation: 5 × 2ˣ⁻¹ = 80
Answer: x = 5
Explanation: Isolate position exponential term, then apply glory logarithm to both sides.