GRAPH OF POLYNOMIAL FUNCTIONS

​GRAPH OF POLYNOMIAL FUNCTION

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Ms. Melo​

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Learning Objectives

In this section, you will:

• Recognize characteristics of graphs of polynomial functions.

• Use factoring to find zeros of polynomial functions.

• Identify zeros and their multiplicities.

• Determine end behavior.

• Understand the relationship between degree and turning points.

• Graph polynomial functions.

• Use the Intermediate Value Theorem.

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Recognizing Characteristics of Graphs of Polynomial Functions

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Solution

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The graphs of ff and hh are graphs of polynomial functions. They are smooth and continuous.

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The graphs of gg and kk are graphs of functions that are not polynomials. The graph of function gg has a sharp corner. The graph of function kk is not continuous.

​Using Factoring to Find Zeros of Polynomial Functions

Recall that if f is a polynomial function, the values of xx for which f(x)=0f(x)=0 are called zeros of f.f. If the equation of the polynomial function can be factored, we can set each factor equal to zero and solve for the zeros.

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We can use this method to find x-x- intercepts because at the x-x- intercepts we find the input values when the output value is zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and fourth-degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials. Consequently, we will limit ourselves to three cases in this section:

1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring.

2. The polynomial is given in factored form.

3. Technology is used to determine the intercepts.

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HOW TO: Given a polynomial function f,f, find the x-intercepts by factoring.

1. Set f(x)=0.f(x)=0.

2. If the polynomial function is not given in factored form:

   1. Factor out any common monomial factors.

   2. Factor any factorable binomials or trinomials.

3. Set each factor equal to zero and solve to find the x-x- intercepts.

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Finding the x-Intercepts of a Polynomial Function by Factoring:

Find the x-intercepts of 

f(x)=x⁶ −3x⁴ +2x² .

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GRAPH: ​

f(x)=x⁶ −3x⁴ +2x² .

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EXAMPLE 3

Finding the x-Intercepts of a Polynomial Function by Factoring

Find the x- intercepts of 

f(x)=x³−5x²−x+5.

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EXAMPLE 3

GRAPH:

f(x)=x³−5x²−x+5.

​HOW TO

Given a graph of a polynomial function, write a formula for the function.

1. Identify the x-intercepts of the graph to find the factors of the polynomial.

2. Examine the behavior of the graph at the x-intercepts to determine the multiplicity of each factor.

3. Find the polynomial of least degree containing all the factors found in the previous step.

4. Use any other point on the graph (the y-intercept may be easiest) to determine the stretch factor.

​This graph has three x-x- intercepts: x=−3, 2 and 5. The y- intercept is located at (0,−2).

​Graphing Polynomial Functions

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​GRAPHING RATIONAL FRACTIONS

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