Section 3.2 Polynomial Functions and Their Graphs

*Definition of a Polynomial Function

Let n be a nonnegative integer and let be real numbers, with . The function defined by

is called a polynomial function of degree n. The number , the coefficient of the variable to the highest power, is called the leading coefficient.

Polynomial functions of degree 2 or higher have graphs that are smooth and continuous.

Example 1) Find the degree and the leading coefficient of each following function.

*The Leading Coefficient Test

As x increases or decreases without bound, the graph of the polynomial functioneventually rises or falls. The behavior of a graph to the far left or the far right is called end behavior. The end behaviorsare determined by the degree n and the leading coefficient .

  1. For n odd (Opposite behavior at each end):
  1. For n even (Same behavior at each end):

Example 2) Use the Leading Coefficient Test to determine the end behavior of the graph of .

*Zeros of Polynomial Functions

Zeros: the values of x for which , i.e. roots or solutions of .

Each real root of the polynomial appears as an x-intercept of the graph.

Example 3) Find all zeroes of .

Example 4) Find all zeroes of .

*Multiplicities of Zeros

In factoring the equation for the polynomial function f, if the same factor occurs times, but not k+1 times, we call ra zero with multiplicity k,i.e. appears.

Example 5) Find the zeros and each multiplicity of.

If r is a zero of even multiplicity, then the graph touches the x-axis and turn around at r.

If r is a zero of odd multiplicity, then the graph crosses the x-axis at r.

Graphs tend to flatten out at zeros with multiplicity greater than one.

Even MultiplicityOdd Multiplicity

Example 6) Find the zeros of and give the multiplicity of each zero. State whether the graph crosses the x-axis or touch the x-axis and turns around at each zero.

*Turning Points of Polynomial Functions

If is a polynomial function of degree n, then the graph of has at most turning points.

*A Strategy for Graphing Polynomial Functions

Step1 Use Leading Coefficient Test to determine the end behavior.

Step2 Find x-intercepts by setting. For a factor of

  1. If k is even, the graph touches the x-axis at r and turns around
  2. If k is odd, the graph crosses the x-axis at r.
  3. If k1, the graph flattens out at (r, 0).

Step 3 Find y-intercept by computing.

Step 4 Use symmetry, if applicable, to help draw the graph:

  1. y-axis symmetry:
  2. Origin symmetry:

Step 5 Use the fact that the maximum number of turning points of the graph is to check whether it is drawn correctly.

Example 7) Graph

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