Parabolas, Vertices, and Transformations

Algebra Foundation

Algebra 1

Study expressions, equations, inequalities, functions, linear models, systems, polynomials, quadratics, and exponential growth with original explanations and worked examples.

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Algebra 1 · Unit 9 · Topic 9.2

Parabolas, Vertices, and Transformations

A quadratic function can be written y = a(x − h)² + k with a ≠ 0.

Definition and explanation

A quadratic function can be written y = a(x − h)² + k with a ≠ 0. Its vertex is (h, k), its symmetry axis is x = h, and the sign of a determines whether it opens upward or downward.

Worked example

For y = −2(x − 3)² + 8, the vertex is (3, 8) and the parabola opens downward. Inputs 2 and 4 both give y = 6. The maximum output is 8 at x = 3 over the real domain.

Common mistake

The vertex's x-coordinate is 3 in (x − 3)², not −3.

Key takeaway

A quadratic function can be written y = a(x − h)² + k with a ≠ 0.

Explain each step of the worked example, then check the units, signs, and conditions that apply.

Further reading

Topic coverage was checked against the openly available OpenStax Elementary Algebra 2e and Algebra and Trigonometry 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.