# Polynomial Functions and Their Graphs

Definition of a Polynomial Function

Let n be a nonnegative integer and let
be

real numbers, with a^{n} ≠ 0 .

The function defined by is called

a polynomial function of x of degree n. The number an , the

coefficient of the variable to the highest power, is called the

leading coefficient.

**Note:** The variable is only raised to positive integer powers–no

negative or fractional exponents.

However, the coefficients may be any real numbers, including

fractions or irrational numbers like π or
.

** Graph Properties of Polynomial Functions**

Let P be any nth degree polynomial function with real coefficients.

The graph of P has the following properties .

1. P is continuous for all real numbers, so there are no breaks,

holes, jumps in the graph.

2. The graph of P is a smooth curve with rounded corners and no

sharp corners.

3. The graph of P has at most n x-intercepts.

4. The graph of P has at most n – 1 turning points.

**Example 1:** Given the following polynomial functions, state the

leading term, the degree of the polynomial and the leading

coefficient.

End Behavior of a Polynomial

Odd-degree polynomials look like y = ±x^{3}.

Even-degree polynomials look like y = ±x^{2} .

** Power functions :**

A power function is a polynomial that takes the form f(x) = ax^{n} ,

where n is a positive integer . Modifications of power functions can

be graphed using transformations.

Even-degree power functions: | Odd-degree power functions: |

**Note:** Multiplying any function by a will multiply all the y-values

by a. The general shape will stay the same. Exactly the same as it

was in section 3.4.

Zeros of a Polynomial

**Example 1:**

Find the zeros of the polynomial and then sketch the graph.

P(x) = x^{3} − 5x^{2} + 6x

If f is a polynomial and c is a real number for which f (c) = 0 , then c

is called a zero of f, or a root of f.

If c is a zero of f, then

• c is an x- intercept of the graph of f.

• (x − c) is a factor of f .

So if we have a polynomial in factored form, we know all of its xintercepts.

• every factor gives us an x-intercept.

• every x-intercept gives us a factor.

**Example 2:** Consider the function

f(x) = −3x(x − 3)^{4}(5x − 2)(2x −1)^{3}(4 − x)^{2}.

**Zeros (x-intercepts):**

**To get the degree, add the multiplicities of all the factors:**

**The leading term is :**

** Steps to graphing other polynomials:**

1. Factor and find x-intercepts.

2. Mark x-intercepts on x-axis.

3. Determine the leading term.

• Degree: is it odd or even?

• Sign : is the coefficient positive or negative?

4. Determine the end behavior. What does it “look like”?

Odd Degree Sign (+) |
Odd Degree Sign (-) |

Even Degree Sign (+) |
Even Degree Sign (-) |

5. For each x-intercept, determine the behavior.

• Even multiplicity: touches x-axis, but doesn’t cross

(looks like a parabola there ).

• **Odd multiplicity of 1:** crosses the x-axis (looks
like a

line there ).

• **Odd multiplicity ≥ 3 :** crosses the x-axis and
looks like a

cubic there .

Note: It helps to make a table as shown in the examples
below.

6. Draw the graph, being careful to make a nice smooth

curve with no sharp corners.

**Note:** without calculus or plotting lots of points, we don’t have

enough information to know how high or how low the turning

points are.

**Example 3:**

Find the zeros then graph the polynomial. Be sure to label the x

intercepts, y intercept if possible and have correct end behavior.

**Example 4:**

Find the zeros then graph the polynomial. Be sure to label the x

intercepts, y intercept if possible and have correct end behavior.

**Example 5:**

Find the zeros then graph the polynomial. Be sure to label the x

intercepts, y intercept if possible and have correct end behavior.

**Example 6:**

Find the zeros then graph the polynomial. Be sure to label the x

intercepts, y intercept if possible and have correct end behavior.

**Example 7:**

Given the graph of a polynomial determine what the equation of

that polynomial.

**Example 8:**

Given the graph of a polynomial determine what the equation of

that polynomial.

Prev | Next |