Quadratic functions

Interpret vertices, symmetry, and change, and find extrema under given conditions.

Meaning

A quadratic function is with and its graph is a parabola. Vertex form makes its symmetry axis and maximum or minimum easier to read.

Example

has vertex and axis Since a square is nonnegative, its minimum over all real numbers is 1.Inputs 2+t and 2−t both give t²+1, so the graph is symmetric about x=2. At the vertex the square is zero, giving the smallest output, 1.An upward-opening parabola has a minimum but no maximum on the reals. On a restricted domain, also check endpoints.

Practice for this concept

Scope and references

Topics

Parabolas, vertices, axes of symmetry, translations, maxima and minima

Learning goal

Interpret vertices, symmetry, and change, and find extrema under given conditions.

Helpful background

Connect coordinates, quadratic values, and equation roots.