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Quizzes > Mathematics > Algebra

Conquer Algebra II: Take the Free Quiz Now

Ready for an Algebra II test online? Challenge your quadratic & exponential skills!

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 II quiz helps you practice quadratic, exponential, logarithmic, and rational functions so you can spot gaps before a test. Work through real-world problems and get instant feedback as you go. Want more? Try a longer practice test or a quick warm-up quiz to keep skills sharp.

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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