Graphing polynomials

To graph a polynomial we need to know three things

1) Type of polynomial

2) Roots

3) y-intercept

Types of Polynomials

Positive odd Starts down ends up

Negative odd Starts up ends down

Positive even Up on both ends

Negative even Down on both ends

We can find the type from the sign and power of the leading term.

The sign tells us if it is positive or negative and the power (degree) tells us if it is even or odd.

Examples:

6 33 4 3 1y x x x

The coefficient is positive

The degree is even

Type: positive even

State the type for each.

4 3 22 3 20y x x x x 5 310 3 2 4y x x x

Negative even

Negative odd

( 5)( 3)(2 4)y x x x 3( )(2 ) 2x x x x

Positive odd

2( 4) (3 1)y x x x 2 4( ) (3 ) 3x x x x

Positive even

2 3(3 )( 4)y x x x 2 3 6( )( )x x x x

Negative even

Roots

The degree of the polynomial, tells us how many roots we will have.

4 3 22 3 20y x x x x 5 310 3 2 4y x x x

2( 4) (3 1)y x x x

2 3(3 )( 4)y x x x

Example: How many roots will the following functions have?

4 roots

5 roots

2 4( ) (3 ) 3x x x x

2 3 6( )( )x x x x

4 roots

6 roots

Roots

Multiplicity of roots

Single root Example: Factor (x+2)

root: -2

graph goes straight through root

double root Example: Factor (x-1)2

root: 1 (M2)

graph goes through the root like a quadratic Indicates a double root

triple root Example: Factor (x+4)3

root: -4 (M3)

graph goes through the root like a cubic Indicates a triple root

RootsMultiplicity of roots

Roots can have higher multiplicity as well.

The graph goes through roots with even multiplicity (M4, M6, M8, etc.) like a quadratic only as the multiplicity increases the graph looks flatter at the root.

Example:

If the root is 3 (M4) If the root is 3 (M12)

Roots

Multiplicity of roots

The graph goes through roots with odd multiplicity (M5, M6, M9, etc.) like a cubic only as the multiplicity increases the graph looks flatter at the root.

Example:

If the root is 3 (M5) If the root is 3 (M11)

Roots

Real roots are x-intercepts.

To find the roots, we let y = 0 and solve for x.

Example: Find the roots for the following.

( 5)( 3)(2 4)y x x x 2( ) ( 4) (3 1)g x x x x

2 3(3 )( 4)y x x x

Roots: -5, 3, -2

Roots: 0, 4 (M2), 1

3

Roots: 0 (M2), 3, -4 (M3)

2( ) ( 1)( 7)f x x x Roots: 1, 7, 7

4 2( 5) ( 2 2)y x x x Roots: 5 (M4), -1+i, -1- i

Y-intercept

To find the y-intercept let x = 0

Always write the y-intercept as a point.

Example: Find the y-intercept for each

( 5)( 3)(2 4)y x x x

2 3(3 )( 4)y x x x

y-intercept: (0, -60)

4 3 22 3 20y x x x x y-intercept: (0, 20)

y-intercept: (0,0)

Now we are ready to graph.

State the type, roots, y-intercept and graph.

2( 3)( 4)( 1)y x x x

Type: ______________________

roots: ______________________

y-intercept: __________

positive odd

3, -4, 1

(0, 24)

First, plot the roots and label

Next, plot and label the y-intercept

Last, sketch the graph (remember the type helps us with the shape)

State the type, roots, y-intercept and graph.

2( 3) (2 1)y x x x

Type: ______________________

roots: ______________________

y-intercept: __________

Negative even

0, -3 (M2),

(0, 0)

1

2

-3 is a double root, so it the graph looks like a quadratic here

State the type, roots, y-intercept and graph.

31(2 )( 4) ( 1)3

y x x x

Type: ______________________

roots: ______________________

y-intercept: __________

positive odd

2, -4 (M3), -1

1280,

3

128

3128

3

-3 is a triple root, so it the graph looks like a cubic here

State the type, roots, y-intercept and graph.

2( 3)( 3)y x x x

Type: ______________________

roots: ______________________

y-intercept: __________

State the type, roots, y-intercept and graph.

3 22(3 ) ( 2)y x x

Type: ______________________

roots: ______________________

y-intercept: __________

State the type, roots, y-intercept and graph.

Type: ______________________

roots: ______________________

y-intercept: __________

Graphing Polynomials Day 2

State the type, roots, y-intercept and graph.

5 4 3 21 1 5 14 2

2 2 2 2y x x x x x

Type: ______________________

roots: ______________________

y-intercept: __________

Expanded Form

Partially Factored 2 214 4 1 2 1

2y x x x x x

Factored form: ______________________________________________ 2 31 1

( 2)( 2) 1 ( 1)( 1) ( 2) ( 1)2 2

y x x x x x x x

Positive odd

-2 (M2), 1 (M3)

(0, -2)

Note: the type and y-intercept are both easy to find from the expanded form.

Sometimes we need to factor in order to find the roots.

State the type, roots, y-intercept and graph.

3 22 10y x x

Type: ______________________

roots: ______________________

y-intercept: __________

Remember to factor first

State the type, roots, y-intercept and graph.

2 3( ) ( 3 )( 2) ( 8 16)f x x x x x x

Type: ______________________

roots: ______________________

y-intercept: __________

State the type, roots, y-intercept and graph.

2 2 4( ) (2 7 4)( 5)( 3)f x x x x x

Type: ______________________

roots: ______________________

y-intercept: __________

Sketch a graph for each description.

Remember real roots are x-intercepts, but imaginary roots are not.

A negative odd function with 5 roots.A positive even function with no real roots

Imaginary roots always come in pairs.

A negative even function with 3 real roots.