6.2 Graphs of Polynomials

The Degree of Polynomials

The degree of a polynomial is the value of the largest exponent.

y = 5x4 + 3x2 – 7

Degree = 4

y = -7x5 + 2x9 + 3x2 – 5x4 + 2

Degree = 9

The Leading Coefficient of a Polynomial

The leading coefficient of a polynomial is the value of coefficient of the term with the largest exponent.

y = 5x4 + 3x2 – 7

Leading Coefficient = 5

y = -7x5 + 2x9 + 3x2 – 5x4 + 2

Leading Coefficient = 2

End Behavior of Polynomials

Use your calculator to graph the following:

y = x

End Behavior of Polynomials

Use your calculator to graph the following:

y = x2

End Behavior of Polynomials

Use your calculator to graph the following:

y = x3

End Behavior of Polynomials

Use your calculator to graph the following:

y = x4

End Behavior of Polynomials

Use your calculator to graph the following:

y = x5

End Behavior of Polynomials

Use your calculator to graph the following:

y = x6

Is there a pattern to the graphs?

y = x6

y = x

y = x2

y = x3

y = x4

y = x5

Graphs of Polynomials with Odd Degreesy = x y = x3 y = x5

y = -x5y = -x y = -x3

Odd degree polynomials are like lines. They start low and end high

Unless they have a negative leading coefficient. Then they start high and end low.

Graphs of Polynomials with Even Degreesy = x6

y = x2 y = x4

y = -x2 y = -x4 y = -x6

Even degree polynomials are like parabolas. They start high and end high

Unless they have a negative leading coefficient. Then they start low and end low.

Roots of PolynomialsWhat are the roots of y = 2 (x - 5)(x - 2)?

What is its degree?

Roots of Polynomials

What are the roots of y = 2 (x - 5)(x - 2)(x + 4)?What is its degree?

Roots of Polynomials

What are the roots of y = -5 (x - 5)(x - 2)(x + 4)(x + 7)?

What is its degree?

What about the leading coefficient?

Roots of Polynomials

What are the roots of y = 3 (x - 5)(x - 2)2?What is its degree?

What about the leading coefficient?

Roots of Polynomials

What are the roots of y = (x - 5)2(x - 2)2?

What is its degree?

What about the leading coefficient?

Roots of Polynomials

What are the roots of y = (x + 3)(x - 5)2(x - 2)2?

What is its degree?

What about the leading coefficient?

Roots of PolynomialsWhat are the roots of y = -(x + 3)3(x - 5)?

What is its degree?

What about the leading coefficient?

Roots of PolynomialsWhat are the roots of y = -(x + 3)3(x - 5)2(x + 7)3?What is its degree?

What about the leading coefficient?

Graphs of Polynomials

To graph a polynomial in factored form:

1.) Determine the degree

2.) Check the sign of the leading coefficient.

3.) Mark on your graph where the curve begins and ends (the end behavior)

4.) Plot the position of the roots.

5.) Connect the dots.

When connecting the dots, don’t forget to check the degree of each root.

Single Roots = Straight through the x-axis (like a line).

Double Roots = Bounce off the x-axis (like a parabola).

Triple Roots = The line bends as it passes through the x-axis (like a cubic).

Graphs of PolynomialsGraph: y = -x(x + 3)5(5x - 10)2(x + 7)3(x - 6)

Graphs of Polynomials

Graph: y = 15x6(x + 8)5(2x - 4)2(3x + 6)3(x - 2)

Graphs of Polynomials

Graph: y = -x2(x + 6)2(x - 4)2(3x + 6)167(x - 8)

Can you figure out a possible equations for this graph?

Can you figure out a possible equations for this graph?

Can you figure out a possible equations for this graph?

Can you figure out a possible equations for this graph?