Section 2.2

Quadratic Functions

It’s Just a Parabola!

• A quadratic function is written in the form

cbxaxxf 2)(

Quadratic functions can also be written in the form

khxaxf 2)()(

The Graph of a Quadratic Function

• The graph of every quadratic function, regardless of how the function is written, is a transformation of the graph

2xy The name of this graph is a parabola.

Concavity

• Concavity is defined as which way the graph opens.

• The graph of a quadratic function is:

• Concave up if a > 0

• Concave down if a < 0

Concavity Examples

32)( 2 xxxf 352)( 2 xxxf

Objectives

• For a given quadratic function, written in either form, we want to be able to find:

1.The vertex

2.The axis of symmetry

3.Domain and range (use interval notation)

4.Intercepts (x and y)

5.Relative extrema (location and value)

6.Sketch the graph

Examples

8)4(2)(

208)(2

2

xxf

xxxf

The Vertex

• The vertex is the lowest point when the parabola is concave up, and the highest point when the parabola is concave down.

• Finding the vertex is extremely important.

cbxaxxf 2)(

• The x-coordinate of the vertex is

a

bx

2

•The y-coordinate of the vertex is found by substituting the x-value into f(x).

a

bfy

2

• The vertex of the parabola is (h,k).

khxaxf 2)()(

The Axis of Symmetry

• The axis of symmetry is an imaginary vertical line that passes through the vertex.

• The equation of the axis of symmetry is

• x = the x-coordinate of the vertex

Domain and Range

• The domain of every quadratic function is

),( •The range of a quadratic function depends on if the graph is concave up or concave down:

Range of a quadratic function

• If the graph is concave up, the range is

),[ y•If the graph is concave down, the range is

],( ywhere y is the y-coordinate of the vertex.

Intercepts of a Quadratic Function

• To find the x-intercept of a quadratic function, substitute 0 for f(x) and solve for x (be prepared to factor).

• To find the y-intercept of a quadratic function, substitute 0 for x and solve for f(x).

Maximums and Minimums

• If your parabola is concave up, the function has a minimum.

• The location of the minimum is the x-coordinate of the vertex.

• The value of the minimum is the y-coordinate of the vertex.

Maximums and Minimums

• If your parabola is concave down, then the function has a maximum.

• The location of the maximum is the x-coordinate of the vertex.

• The value of the maximum is the y-coordinate of the vertex.

Application Problems

• In most applications, you will be asked to find a maximum or minimum value for a given quadratic function.

• Recall that a maximum or a minimum will be the y-coordinate of the vertex.

• If the function graph is concave up (a > 0), the function will have a minimum.

• If the function graph is concave down

(a < 0), the function will have a maximum.

More…

• Finding your quadratic function may require algebra and geometry.

• When in doubt, try drawing a picture or make a table to better illustrate what is going on with the problem.

Example 1

An athlete whose event is the shot put releases a shot. The height, f(x), in feet, can be modeled by f(x)=-0.01x2 + 0.6x + 6.3, where x is the shot’s horizontal distance, in feet, from its point of release. What is the maximum height of the shot and how far from its point of release does this occur?

Example 2

Among all pairs of numbers whose sum is 20, find a pair whose product is as large as possible. What is the maximum product?

Example 3

Farmer Ed has 300 meters of fencing, and wants to enclose a rectangular plot that borders on a river. If Farmer Ed does not fence the side along the river, find the length and the width of the plot that will maximize the area. What is the largest area that can be enclosed?

Example 4

On a certain route, an airline carries 8000 passengers per month, each paying $70. A market survey indicates that for each $1 increase in the ticket price, the airline will lose 100 passengers. Find the ticket price that will maximize the airline’s monthly revenue for the route. What is the maximum monthly revenue?