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Use the graph of the function of degree 5 in [link] to identify the zeros of the function and their multiplicities.

Graph of an even-degree polynomial with degree 6.

The graph has a zero of –5 with multiplicity 1, a zero of –1 with multiplicity 2, and a zero of 3 with even multiplicity.

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Determining end behavior

As we have already learned, the behavior of a graph of a polynomial function    of the form

f ( x ) = a n x n + a n 1 x n 1 + ... + a 1 x + a 0

will either ultimately rise or fall as x increases without bound and will either rise or fall as x decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say –100 or –1,000.

Recall that we call this behavior the end behavior of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, a n x n , is an even power function, as x increases or decreases without bound, f ( x ) increases without bound. When the leading term is an odd power function, as x decreases without bound, f ( x ) also decreases without bound; as x increases without bound, f ( x ) also increases without bound. If the leading term is negative, it will change the direction of the end behavior. [link] summarizes all four cases.

Graph of a polynomial function with degree 5.

Understanding the relationship between degree and turning points

In addition to the end behavior, recall that we can analyze a polynomial function’s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function f ( x ) = x 4 x 3 4 x 2 + 4 x in [link] . The graph has three turning points.

Graph of an odd-degree polynomial with a negative leading coefficient. Note that as x goes to positive infinity, f(x) goes to negative infinity, and as x goes to negative infinity, f(x) goes to positive infinity.

This function f is a 4 th degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.

Interpreting turning points

A turning point    is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).

A polynomial of degree n will have at most n 1 turning points.

Finding the maximum number of turning points using the degree of a polynomial function

Find the maximum number of turning points of each polynomial function.

  1. f ( x ) = x 3 + 4 x 5 3 x 2 + 1
  2. f ( x ) = ( x 1 ) 2 ( 1 + 2 x 2 )
  1. First, rewrite the polynomial function in descending order: f ( x ) = 4 x 5 x 3 3 x 2 + 1

    Identify the degree of the polynomial function. This polynomial function is of degree 5.

    The maximum number of turning points is 5 1 = 4.

  2. First, identify the leading term of the polynomial function if the function were expanded.

    Graph of f(x)=x^4-x^3-4x^2+4x which denotes where the function increases and decreases and its turning points.

    Then, identify the degree of the polynomial function. This polynomial function is of degree 4.

    The maximum number of turning points is 4 1 = 3.

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Graphing polynomial functions

We can use what we have learned about multiplicities, end behavior, and turning points to sketch graphs of polynomial functions. Let us put this all together and look at the steps required to graph polynomial functions.

Given a polynomial function, sketch the graph.

  1. Find the intercepts.
  2. Check for symmetry. If the function is an even function, its graph is symmetrical about the y - axis, that is, f ( x ) = f ( x ) . If a function is an odd function, its graph is symmetrical about the origin, that is, f ( x ) = f ( x ) .
  3. Use the multiplicities of the zeros to determine the behavior of the polynomial at the x - intercepts.
  4. Determine the end behavior by examining the leading term.
  5. Use the end behavior and the behavior at the intercepts to sketch a graph.
  6. Ensure that the number of turning points does not exceed one less than the degree of the polynomial.
  7. Optionally, use technology to check the graph.

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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