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Algebra II Readiness and Final Test

A free Algebra II lesson from the “Trigonometry and Modeling” unit, with a worked example and practice problems including step-by-step solutions.

The Algebra II readiness test samples function notation, domain and range, transformations, quadratics, polynomial tools, rational expressions, radicals, exponential functions, and logarithms.

What you'll learn

Why it matters: The Algebra II readiness test mirrors what college placement exams, ACT/SAT math, and pre-calculus diagnostics expect. The skill is identifying which Algebra II tool each problem is asking for — equations, functions, polynomials, exponentials, logs, or trig — and applying it cleanly.

Worked example

Problem. Evaluate log base 2 of 16.

  1. Ask: 2 to what power equals 16?
  2. 2^4 = 16.
  3. So the logarithm is 4.

Answer: 4

Practice problems

1. If f(x) = 3x + 2, find f(8).

Show solution
  1. Course Review: First identify exactly what the question is asking: If f(x) = 3x + 2, find f(8).
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. 3(8) + 2 = 26.
  4. Check the result by substituting or estimating: the response should match 26 and make sense in the original problem.

Answer: 26

2. For f(x) = 1/(x + 4), what value is excluded?

Show solution
  1. Course Review: First identify exactly what the question is asking: For f(x) = 1/(x + 4), what value is excluded?
  2. For function notation, treat the value inside parentheses as the input and carefully substitute it into the rule.
  3. x + 4 cannot be zero.
  4. Check the result by substituting or estimating: the response should match -4 and make sense in the original problem.

Answer: -4

3. The vertex of y = (x - 6)^2 - 1 is...

Choices: (6, -1) · (-6, -1) · (6, 1) · (-6, 1)

Show solution
  1. Course Review: First identify exactly what the question is asking: The vertex of y = (x - 6)^2 - 1 is...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Vertex form gives (h, k).
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (6, -1)

4. Factor x^2 - 7x + 12.

Choices: (x - 3)(x - 4) · (x + 3)(x + 4) · (x - 2)(x - 6) · (x - 1)(x - 12)

Show solution
  1. Course Review: First identify exactly what the question is asking: Factor x^2 - 7x + 12.
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. 3 and 4 multiply to 12 and add to 7.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 3)(x - 4)

5. Find the discriminant of x^2 - 6x + 9 = 0.

Show solution
  1. Course Review: First identify exactly what the question is asking: Find the discriminant of x^2 - 6x + 9 = 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 36 - 36 = 0.
  4. Check the result by substituting or estimating: the response should match 0 and make sense in the original problem.

Answer: 0

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