Introduction

Equation and Graph

P(x) = x3 − 4x2 − 7x + 10 is graphed here

Notice there are two turning points in the graph. These are called critical points.

Also notice the graph crosses the x-axis three times, so there are three x-intercepts.

A graph of a polynomial that starts down and ends up with 2 turning points and 3 x-intercepts.

Equation and Graph 2

Look at each equation and graph to see the number of turning points and x-intercepts. Is there a connection that you can make between the degree and the turning points and x-intercepts?

P(x) = x2 + 7x + 6

Number of turning points: 1

Number of x-intercepts: 2

A graph of a polynomial that starts up and ends up with 1 turning point and 2 x-intercepts.

Equation and Graph 3

Look at each equation and graph to see the number of turning points and x-intercepts. Is there a connection that you can make between the degree and the turning points and x-intercepts?

P(x) = x3 + 4x2 + 7x + 10

Number of turning points: 2

Number of x-intercepts: 3

A graph of a polynomial that starts down and ends up with 2 turning points and 3 x-intercepts.

Equation and Graph 4

Look at each equation and graph to see the number of turning points and x-intercepts. Is there a connection that you can make between the degree and the turning points and x-intercepts?

P(x) = x3 + 5x2 + 5x + 12

Number of turning points: 2

Number of x-intercepts: 1

A graph of a polynomial that starts down and ends up with 2 turning points and 1 x-intercept.

Equation and Graph 5

Look at each equation and graph to see the number of turning points and x-intercepts. Is there a connection that you can make between the degree and the turning points and x-intercepts?

P(x) = 2x4 − 13x3 − 19x2 − 10x + 24

Number of turning points: 3

Number of x-intercepts: 4

A graph of a polynomial that starts up and ends up with 3 turning points and 4 x-intercepts.

Equation and Graph 6

Look at each equation and graph to see the number of turning points and x-intercepts. Is there a connection that you can make between the degree and the turning points and x-intercepts?

P(x) = x5 − 5x4 + 20x2 − 16x

Number of turning points: 4

Number of x-intercepts: 5

A graph of a polynomial that starts down and ends up with 4 turning points and 5 x-intercepts.