Find the vertex, intercepts, and symmetry axis of a parabola.

Write a quadratic function in vertex form.

Solve application problems involving quadratic functions and parabolas.

Quadratic functions

A quadratic function is a 2nd degree polynomial function. Every quadratic function can be written in the form , where , , and are real numbers with .

The graph of a quadratic function is a smooth U-shaped curve called a parabola. The turning point at the tip of the U is called the vertex, and the graph is symmetric about the vertical line through the vertex.

Any quadratic function can be written in the equivalent vertex form

by using a process called completing the square.

The vertex is at the point , where and .

The axis of symmetry is the vertical line .

The parabola opens upward if and downward if .

The graph is wider than the graph of if , as wide if , and narrower if .

The -intercepts of the graph can be found by solving (i.e., finding the zeros of ). If a parabola has two distinct -intercepts, the vertex is directly between them.

The -intercept is .

The quadratic function attains an extreme value (max or min) at the vertex of its graph.

Examples

Consider the parabola described by . Find the -intercepts, the coordinates of the vertex, and the axis of symmetry. Finally, write the equation in vertex form and sketch the graph.

Sketch the graph of . Describe the features of the graph/function.

A ball is thrown upward with a velocity of feet per second from the top of a -foot building. What is the maximum height of the ball? How long until the ball hits the ground?

A quadratic function has a leading coefficient of and zeros and . Write its equation in standard form and vertex form.