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Quizzes > Mathematics

Algebra 2 Quiz: Quadratics, Exponents, Logs, and Rational Functions

Quick, free Algebra II practice test with instant results.

Editorial: Review CompletedCreated By: Joop MeijerUpdated Aug 27, 2025
Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art of algebra graphs on teal background for Algebra2 quiz on quadratic exponential logarithmic and rational functions

This Algebra 2 quiz helps you practice quadratic, exponential, logarithmic, and rational functions, so you can check understanding and spot gaps before a test. For extra review, try an algebra 2 practice test, build fluency with simplifying expressions practice, or focus on equations in the solve for x quiz.

What is the value of the discriminant for the quadratic function f(x) = 2x^2 - 4x + 1?
0
8
4
-8
The discriminant of a quadratic ax^2+bx+c is b^2 - 4ac. Here, b = -4 and a = 2, c = 1, so (-4)^2 - 4*2*1 = 16 - 8 = 8. A positive discriminant indicates two distinct real roots. For more details, see .
What is the vertex of the parabola defined by h(x) = (x + 3)^2 - 5?
(-3, -5)
(5, -3)
(-3, 5)
(3, -5)
A parabola in the form (x - h)^2 + k has its vertex at (h, k). Here the expression is (x - (-3))^2 - 5, so the vertex is at (-3, -5). This shift comes from moving 3 units left and 5 units down. See for more.
Evaluate f(2) for the exponential function f(x) = 3 * 2^x.
12
6
9
8
Substitute x = 2 into f(x)=3*2^x to get f(2)=3*2^2=3*4=12. Exponential functions scale rapidly as x increases. For more on evaluating exponentials, see .
Solve the quadratic equation x^2 - 5x + 6 = 0.
x = -2 or x = -3
x = -1 or x = -6
x = 1 or x = 6
x = 2 or x = 3
Factor x^2 - 5x + 6 into (x - 2)(x - 3) = 0, so x = 2 or x = 3. Factoring is often the quickest method when integer roots exist. For strategies, see .
Simplify the expression log(x) + log(4) - log(2).
log(8x)
log(4x)
log(2x)
log(x/2)
Combine log(x) + log(4) into log(4x), then subtract log(2) giving log((4x)/2) = log(2x). Logarithm properties allow summation and subtraction in this way. See for details.
What are the vertical asymptotes of the rational function f(x) = (x^2 - 1) / (x^2 - 4)?
x = 2 only
x = 1 and x = -1
x = -2 only
x = 2 and x = -2
Vertical asymptotes occur where the denominator is zero and the factor does not cancel with the numerator. Here x^2 - 4 = (x - 2)(x + 2), which never cancels, so vertical asymptotes are x = 2 and x = -2. For more, see .
Solve the equation 5^x = 125.
1/3
2
125
3
Recognize that 125 = 5^3, so 5^x = 5^3 implies x = 3. When bases match, exponents must be equal. See for more examples.
Determine the domain of the function f(x) = ?(x - 2) / (x^2 - 9).
[2, ?)
(2, ?)
[2, 3) ? (3, ?)
(??, 2]
The square root requires x ? 2 ? 0, so x ? 2, and the denominator x^2 - 9 ? 0, so x ? ±3. Combined, the domain is [2, 3) ? (3, ?). For domain rules, see .
In the function y = ?2·3^(x ? 4) + 5, how is the graph transformed compared to y = 3^x?
Shifted left 4, up 5, compressed by factor 2
Reflected over the y-axis, shifted right 4, down 5
Reflected over the x-axis, vertically stretched by factor 2, shifted right 4 and up 5
Shifted right 4, down 5, stretched by factor 3
Starting with y = 3^x, replacing x with x?4 shifts right 4 units, multiplying by ?2 reflects over the x-axis and stretches by |2|, and adding 5 shifts the graph up 5 units. Transformations are applied in the order of operations. See for details.
Solve for x: log_2(x ? 1) + log_2(x + 1) = 3.
3
?3
1
x = 3 or x = ?3
Combine logs: log_2((x?1)(x+1)) = 3 implies (x^2 ? 1) = 2^3 = 8, so x^2 = 9. Solutions are x = ±3, but the arguments of logs require x > 1, so only x = 3 is valid. See .
Find the inverse function of f(x) = ln((x ? 2)/3) + 4.
f?¹(x) = 3·e^(x ? 4) + 2
f?¹(x) = e^(x ? 4)/3 + 2
f?¹(x) = e^(x + 4) + 2
f?¹(x) = 3·ln(x ? 4) + 2
To find the inverse, set y = ln((x?2)/3) + 4, subtract 4: y?4 = ln((x?2)/3), exponentiate: e^(y?4) = (x?2)/3, then x = 3·e^(y?4) + 2. Replace y with x to get f?¹(x) = 3e^(x ? 4) + 2. For more, see .
0
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Study Outcomes

  1. Solve Quadratic Equations -

    Use the free Algebra II quiz to master solving quadratic functions through factoring, completing the square, and the quadratic formula with instant feedback.

  2. Apply Exponential Function Rules -

    Practice and reinforce exponential equations quiz techniques by evaluating growth and decay models and understanding their applications.

  3. Analyze Logarithmic Relationships -

    Break down logarithmic equations practice to convert between exponential and logarithmic forms and solve real-world problems.

  4. Simplify Rational Expressions -

    Sharpen your skills in reducing, multiplying, and dividing rational functions as featured in this Algebra II test online.

  5. Interpret Function Graphs -

    Read and analyze key features of quadratic, exponential, and logarithmic graphs to predict behavior and identify critical points.

  6. Track and Assess Your Progress -

    Monitor your performance on each quiz section to target areas for improvement and build confidence in Algebra II concepts.

Cheat Sheet

  1. Master the Quadratic Discriminant -

    The discriminant (D = b² − 4ac) tells you if your quadratic has two real, one real, or two complex roots, making it a staple for an Algebra II quiz. For example, D > 0 means two distinct real solutions, D = 0 means one repeated root, and D < 0 indicates complex roots (source: MIT OpenCourseWare). Use this quick check before leaping into any quadratic functions quiz to save time and build confidence.

  2. Convert to Vertex Form -

    Rewriting ax² + bx + c as a(x − h)² + k reveals the vertex (h, k) directly; h = - b/(2a) and k = c - b²/(4a). This technique - often called "completing the square" - is vital for graphing and optimization problems (refer to Khan Academy's Algebra II resources). Mastering vertex form is a game-changer on any Algebra II test online.

  3. Understand Exponential Growth & Decay -

    Exponential functions take the form f(x) = a·bˣ, where b > 1 models growth and 0 < b < 1 models decay, crucial for the exponential equations quiz. Real-world applications include population growth (b > 1) and radioactive half-life (b = ½), giving extra context to practice problems (source: University of California, Irvine). Remember "growth if b goes up, decay if b goes down" as your mnemonic.

  4. Apply Logarithm Laws & Change-of-Base -

    Logarithms are inverses of exponentials: log_b(x·y) = log_b(x) + log_b(y) and log_b(xᶜ) = c·log_b(x). The change-of-base formula log_b(a) = log(a)/log(b) (using any common log) is a must-know for your logarithmic equations practice (source: Paul's Online Math Notes, Lamar University). Solidify these laws to breeze through log sections on your Algebra II test online.

  5. Analyze Rational Function Asymptotes -

    Rational functions, f(x) = P(x)/Q(x), feature vertical asymptotes at Q(x)=0 (if P doesn't cancel) and horizontal asymptotes determined by comparing polynomial degrees (source: Ohio State University). For instance, if deg P < deg Q, y=0; if equal, y = leading coeff ratio. Spotting these behaviors quickly boosts your score on the rational functions portion of an Algebra II quiz.

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