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Example 1 GRAPHING FUNCTIONS OF THE FORM (x) = axn

Solution Choose several values for x, and find the corresponding values of (x), or y.

a.Graph the function.

3( )x xf

x (x)

– 2 – 8– 1 – 10 0

1 1

2 8

3( )x xf

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If the zero has even multiplicity, the graph is tangent to the x-axis at the corresponding x-intercept (that is, it touches but does not cross the x-axis there).

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If the zero has odd multiplicity greater than one, the graph crosses the x-axis and is tangent to the x-axis at the corresponding x-intercept. This causes a change in concavity, or shape, at the x-intercept and the graph wiggles there.

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Turning Points and End Behavior

The previous graphs show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.

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Turning Points

A polynomial function of degree n has at most n – 1 turning points, with at least one turning point between each pair of successive zeros.

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End Behavior

The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree. A polynomial of the form

11 0( ) n n

n nx a x a x a f

has the same end behavior as . ( ) nnx a xf

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End Behavior of Polynomials

Suppose that axn is the dominating term of a polynomial function of odd degree.1.If a > 0, then as and as Therefore, the end behavior of the graph is of the type that looks like the figure shown here.

We symbolize it as .

, ( ) ,x x f, ( ) .x x f

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End Behavior of Polynomials

Suppose that axn is the dominating term of a polynomial function of odd degree.2. If a < 0, then as and as

Therefore, the end behavior of the graph looks like the graph shown here.

We symbolize it as .

, ( ) ,x x f, ( ) .x x f

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End Behavior of Polynomials

Suppose that axn is the dominating term of a polynomial function of even degree.1.If a > 0, then as Therefore, the end behavior of the graph looks like the graph shown here.

We symbolize it as .

, ( ) .x x f

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End Behavior of Polynomials

Suppose that is the dominating term of a polynomial function of even degree.2. If a < 0, then as Therefore, the end behavior of the graph looks like the graph shown here.

We symbolize it as .

, ( ) .x x f

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL

Match each function with its graph.4 2( ) 5 4x x x x f

Solution Because is of even degree with positive leading coefficient, its graph is C.

A. B. C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL

Match each function with its graph.6 2( ) 3 4x x x x g

Solution Because g is of even degree with negative leading coefficient, its graph is A.

A. B. C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL

Match each function with its graph.3 2( ) 3 2 4x x x x h

Solution Because function h has odd degree and the dominating term is positive, its graph is in B.

A. B. C. D.

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Example 3 DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL

Match each function with its graph.7( ) 4x x x k

Solution Because function k has odd degree and a negative dominating term, its graph is in D.

A. B. C. D.

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Graphing Techniques

We have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function will show the following characteristics:1. all x-intercepts (zeros)2. the y-intercept3. the sign of (x) within the intervals formed by the x-intercepts, and all turning points4. enough of the domain to show the end behavior.

In Example 4, we sketch the graph of a polynomial function by hand. While there are several ways to approach this, here are some guidelines.