Section 4.1 - Quadratic Functions and Applications

Section Objectives

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

Write a quadratic function in vertex form.

Find the equation of a quadratic function from its graph.

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 standard 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.

By completing the square, any quadratic function in standard form, , can be rewritten in the equivalent vertex form

.

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 quadratic function defined by . Find the -intercepts of the graph, 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.

The total cost of manufacturing a set of golf clubs is given by

where is the number of sets of golf clubs produced. How many sets of golf clubs should be manufactured to incur minimum cost and what is that minimum cost?

Cindy wants to construct two side-by-side dog-training pens of the same size and sharing one common side. She has 400 ft of fencing material to use. What values of and maximize the combined areas of the pens?

The graph of a parabola is shown below. Find an equation for the graph.