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Quadratic Applications

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

Quadratic applications often ask about maximum or minimum values, when something reaches the ground, or dimensions that create an area. The vertex and zeros carry the meaning.

What you'll learn

Why it matters: Maximum profit, peak projectile height, minimum-cost packaging, and braking-distance problems all reduce to a quadratic optimization. The vertex is the answer, and the zeros tell you when a value crosses a target.

Worked example

Problem. A height model is h(t) = -t^2 + 8t. When does the object hit the ground after launch?

  1. Set h(t) = 0.
  2. Factor -t(t - 8) = 0.
  3. The later time is t = 8.

Answer: 8

Practice problems

1. For h(t) = -t^2 + 10t, when does the maximum occur?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For h(t) = -t^2 + 10t, when does the maximum occur?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The zeros are 0 and 10.
  4. The vertex is halfway at 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

2. For h(t) = -t^2 + 12t, when does the object return to the ground after launch?

Show solution
  1. Warm-up: First identify exactly what the question is asking: For h(t) = -t^2 + 12t, when does the object return to the ground after launch?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Factor -t(t - 12).
  4. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

3. A rectangle has sides x + 2 and x + 8. If x = 4, what is the area?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A rectangle has sides x + 2 and x + 8. If x = 4, what is the area?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The side lengths are 6 and 12.
  4. Check the result by substituting or estimating: the response should match 72 and make sense in the original problem.

Answer: 72

4. A quadratic has zeros 3 and 15. What is the axis of symmetry?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A quadratic has zeros 3 and 15. What is the axis of symmetry?
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Average the zeros.
  4. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

5. A revenue model opens down. Its vertex represents the...

Choices: Maximum revenue · Minimum revenue · Input only · No solution

Show solution
  1. Challenge: First identify exactly what the question is asking: A revenue model opens down. Its vertex represents the...
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. Opening down means the vertex is highest.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Maximum revenue

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