4.3 GRAPHS OF POLYNOMIAL FUNCTIONS

Objectives:

1. Recognize the shape of basic polynomial functions.

2. Describe the graph of a polynomial function.

3. Identify properties of general polynomial functions: Continuity, End Behavior, Intercepts, Local Extrema, Points of Inflection.

4. Identify complete graphs of polynomial functions.

CONTINUITY

A continuous function is an unbroken curve with no jumps, gaps, or holes.

Both of these functions are discontinuous because they demonstrate jumps, gaps, & holes.

A polynomial function is always continuous and will never have a sharp corner. They also will have no asymptotes.

END BEHAVIOR

The end behavior is the shape of a polynomial graph at the far left or right that determines the final direction a function will continue. In other words, the “behavior” of a graph at its “ends.”

EVEN DEGREE POLYNOMIAL FUNCTIONS

Both ends go the same direction.

Positive Leading Coefficient(Up, Up)

Negative Leading Coefficient(Down, Down)

ODD DEGREE POLYNOMIAL FUNCTIONS

The ends go in opposite directions.

Positive Leading Coefficient(Down, Up)

Negative Leading Coefficient(Up, Down)

INTERCEPTS

For any polynomial function written in standard form & degree n, its graph has: Exactly one y-intercept equal to the constant

term. At most n x-intercepts equal to the degree of the

function.

For example, the polynomial:

Has a y-intercept at 4. Has at most four x-intercepts.

f(x) 3x 4 x 3 x 4

MULTIPLICITY The number of times a factor of a polynomial occurs

when the expression is written in completely factored form.

If the multiplicity is even, the graph touches the x-axis & reverses direction, if odd, the graph passes through the x-axis.

EXAMPLE #1MULTIPLICITY OF ZEROS

Find all the zeros of f(x) x 1 3 x 1 x 3 2 .

State the multiplicity of each zero, and state whether

the graph of f touches or crosses the x-axis at each

corresponding x-intercept.

Zero

Multiplicity

Behavior

−1 3 Crosses(Odd

Multiplicity)

1 1 Crosses(Odd

Multiplicity)

3 2 Touches(Even

Multiplicity)

1 2 3 4 5–1–2–3–4–5 x

10

20

30

40

50

–10

–20

–30

–40

–50

y

A graph will confirm.

LOCAL EXTREMA

Local extrema are peaks or valleys in the graph of a polynomial function

POINTS OF INFLECTION

A point of inflection is the point where the concavity of a graph of a function changes.

Basically this is saying that ANY polynomial function with degree greater than or equal to 2 will have at most (n – 2) points of inflection. Odd polynomial functions starting with degree 3 or more will have at least 1.

There is NO guarantee that an even polynomial function will have any points of inflection.

EXAMPLE #2A COMPLETE GRAPH OF A POLYNOMIAL

Find a complete graph of

f(x) x 4 11x 3 25x 2 50x 70.

A.)

Since the degree is even and the leading coefficient positive, the graph to point up at the left and the right.

Additionally, since the degree is 4, it has at most 4 real zeros, at most 3 local extrema (4 – 1 = 3), and at most 2 points of inflection (4 – 2 = 2).

2 4 6 8 10–2–4–6–8–10 x

2

4

6

8

10

–2

–4

–6

–8

–10

y

2 4 6 8 10–2–4–6–8–10 x

20

40

60

80

100

–20

–40

–60

–80

–100

y

The default view is incomplete so we must change the scale on the y-axis to accommodate the complete graph. Remember that −70 is the y-intercept.

EXAMPLE #2A COMPLETE GRAPH OF A POLYNOMIAL

Find a complete graph of

f(x) x 3 3.9x2 5x 1.

B.)

A graph is complete when all zeros & local extrema are visible, and the correct behavior for both ends is shown.For this particular polynomial function, the degree is odd and the leading coefficient is even, this means that the function will point down towards the left and up towards the right.

Additionally, since the degree is 3 it will have at most 3 real zeros, 2 local extrema (3 – 1 = 2), and one point of inflection (since it is odd is has at least one and 3 – 2 = 1 for the maximum number of inflection points).

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

This graph is therefore complete.

EXAMPLE #3DETERMINING A POLYNOMIAL FROM A GRAPH

Determine whether the given graph could possibly represent a polynomial function of degree 3, degree 4, or degree 5?

1 2 3–1–2–3 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

Identify the following:

End Behavior:

Zeros:

Local Extrema:

Points of Inflection:

Down, Up

3

4

3

EXAMPLE #3DETERMINING A POLYNOMIAL FROM A GRAPH

Determine whether the given graph could possibly represent a polynomial function of degree 3, degree 4, or degree 5?

1 2 3–1–2–3 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

End Behavior:Zeros:Local Extrema:Points of Inflection:

Down, Up3

43

Working backwards with the local extrema or inflection points:

n – 1 = 4 n = 5n – 2 = 3 n = 5

This implies that the graph could not be a 3rd or 4th degree polynomial, as well as the fact that it must be odd, but it could be a 5th.

EXAMPLE #4A COMPLETE GRAPH OF A POLYNOMIAL

Determine if the graph shown below

is a complete graph of

f(x) 0.02x 5 x 4 x 3 5x 2 4x 2.

This could not possibly be a complete graph because the end behavior shown is up, up, and would then have an even degree, this polynomial has an odd degree of 5.