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Quadratic Applications

A free College Algebra lesson from the “Quadratic Functions and Equations” unit, with a worked example and practice problems including step-by-step solutions.

Quadratic models often describe height, area, revenue, and optimization. Zeros can represent times or break-even points, while vertices represent maximum or minimum values.

What you'll learn

Why it matters: Maximum height, maximum revenue, minimum cost, and area optimization are quadratic application signals. The vertex answers the best-value question, while zeros answer when the modeled quantity reaches a target such as ground level or break-even.

Worked example

Problem. A height model is h(t) = -t^2 + 10t. When does the object return to the ground?

  1. Set h(t) = 0.
  2. Factor -t(t - 10) = 0.
  3. The later time is t = 10.

Answer: 10

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. For h(t) = -t^2 + 12t, when does the maximum occur?

Show solution
  1. Halfway between the zeros 0 and 12.

Answer: 6

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

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  1. The sides are 5 and 10.

Answer: 50

3. A downward-opening revenue parabola has its maximum at the...

Choices: Vertex · y-intercept only · Left endpoint always · Denominator

Show solution
  1. Opening down means the vertex is highest.

Answer: Vertex

4. For h(t) = -t^2 + 6t, when does the maximum occur?

Show solution
  1. The axis is halfway between zeros 0 and 6.

Answer: 3

5. For R(p) = -2p^2 + 40p, what price gives maximum revenue?

Show solution
  1. Use -b/(2a): -40/(2 x -2) = 10.

Answer: 10

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