Quadratic Functions Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Functions.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A quadratic function is a polynomial function of degree 2, written as f(x) = ax^2 + bx + c with a \neq 0, whose graph is a U-shaped curve called a parabola that opens upward when a > 0 or downward when a < 0.

The path of a thrown ball — rising then falling — traces a parabola opening downward.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Quadratics model acceleration, projectiles, and optimization.

Common stuck point: Vertex form vs standard form—each reveals different information.

Sense of Study hint: When you see a quadratic, first identify a, b, and c in the standard form ax^2 + bx + c. Then find the vertex using x = -b/(2a) and compute y at that point. Finally, determine the direction of opening from the sign of a and plot a few points on either side of the vertex.

Worked Examples

Example 1

easy
Find the vertex of f(x) = x^2 - 6x + 8.

Solution

  1. 1
    The x-coordinate of the vertex is x = -\frac{b}{2a} = -\frac{-6}{2(1)} = 3.
  2. 2
    The y-coordinate is f(3) = 9 - 18 + 8 = -1.
  3. 3
    The vertex is (3, -1).

Answer

(3, -1)
For a quadratic f(x) = ax^2 + bx + c, the vertex formula x = -\frac{b}{2a} gives the axis of symmetry. Plugging this x back in gives the minimum (if a > 0) or maximum (if a < 0) value.

Example 2

medium
Does the parabola g(x) = -2x^2 + 4x + 1 open upward or downward? Find its maximum value.

Example 3

medium
Find the vertex and axis of symmetry of f(x) = 2x^2 - 8x + 3.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the y-intercept of f(x) = 3x^2 - 2x + 5.

Example 2

medium
Find the zeros of f(x) = x^2 - 4x - 5.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

linear functionsexponents