# Graphing Polynomial Functions

How to graph polynomial functions by using its zeros: basic graphs, examples, and their solutions.

## Graphs: Odd Power Functions

The graphs of odd power functions

(*y* = *x*, *y* = *x*^{3}, *y* = *x*^{5}, ...)

pass through the x-axis

at their zeros.

## Graphs: Even Power Functions

The graphs of even power functions

(*y* = *x*^{2}, *y* = *x*^{4}, *y* = *x*^{6}, ...)

bounce off the x-axis

at their zeros.

## Example 1

Use the synthetic division

to factor the right side of the function.*y* = (*x* + 1)(*x* - 2)^{2}

Factor theorem

The zeros are -1 and 2.

Mark the zeros on the *x*-axis: -1, 2.

The sign of the highest degree term is (+). (brown)

As *x* → ∞, *y* → ∞.

So the graph starts at quadrant I. (brown point)

Graph the function from the right to the left.

(*x* - 2)^{2} is an even power factor.

So at *x* = 2,

the graph bounces off the *x*-axis.

[↖ ↙]

(*x* + 1) is an odd power factor.

So at *x* = -1,

the graph passes through the *x*-axis.

[↙ ↙]

## Example 2

Factor the right side of the function.*y* = *x*^{2}(*x* + 1)(*x* - 1)

The zeros are 0, -1, and 1.

Factoring the difference of squares

Mark the zeros on the *x*-axis: -1, 0, 1.

The sign of the highest degree term is (+). (brown)

So the graph starts at the quadrant I. (brown point)

Graph the function from the right to the left.

(*x* - 1) is an odd power factor.

So at *x* = -1,

the graph passes through the *x*-axis.

[↙ ↙]*x*^{2} is an even power factor.

So at *x* = 0,

the graph bounces off the *x*-axis.

[↙ ↖]

(*x* + 1) is an odd power factor.

So at *x* = -1,

the graph passes through the *x*-axis.

[↖ ↖]

## Example 3

The function is already factored.

And the zeros are 0 and -3.

Mark the zeros on the *x*-axis: -3, 0.

The sign of the highest degree term is (-). (brown)

As *x* → ∞, *y* → -∞.

So the graph starts at the quadrant IV. (brown point)

Graph the function from the right to the left.*x*^{3} is an odd power factor.

So at *x* = 0,

the graph passes through the *x*-axis.

[↖ ↖]

(*x* + 3)^{2} is an even power factor.

So at *x* = -3,

the graph bounces off the *x*-axis.

[↖ ↙]