Math 121                        End Behavior and Number of Extrema for Polynomial Functions

End Behavior: Let’s look at a few of examples of polynomial functions. Draw the graphs of f(x) = 2x³ + 3x² -12x - 4 and g(x) = 2x³ on your calculator. Use the following viewing rectangle: xmin = -5, xmax = 5, ymin = -60, ymax = 60. After you look at that graph change the viewing rectangle to: xmin = -8, xmax = 8, ymin = -1000, and ymax = 1000.

The next two graphs are h(x) = -2x⁴ - 24x³ - 96x² -128x and k(x) = -2x⁴.   Try viewing these graphs with the following viewing rectangle: xmin = -7, xmax = 4, ymin = -70, and ymax = 55. Now look at the same graph with xmin = -16, xmax = 14, ymin = -60,000 , and ymax = 100 .

Notice that the end behavior of the first two graphs match each other and that the end behavior of the second two match each other. Why do you think that is? As you plug in large numbers for x the term with the largest exponent will have the greatest effect. This is also true when you plug in large negative numbers.

Thus you can determine the end behavior of a polynomial by looking at its first term ( the term with the highest exponent).

Notice that if you raise a number to an even power you get a positive number and that if you raise a number to an odd power the sign you get matches the sign of the number. There is no need to memorize this. You can figure it out from your current knowledge.

pos^even = pos     neg^even = pos

pos^odd= pos      neg^odd = neg

Polynomial functions always head towards as x goes towards ¥ . All you need to look at is the coefficient of the first term and whether the polynomial has an even or odd degree.

Notice the end behavior of a polynomial of even degree is the same on both sides and the end behavior of a polynomial of odd degree is opposite on the two ends. This brings us to a discussion of the number of turning points (extemum) of the polynomial.

Number of local extrema: A local extremum is a point that is higher or lower than all the points ‘near’ it.

Notice that for an odd function there is an even number of extrema and for an even function there is an odd number of extrema. Also notice that for a function of degree n the most extrema we can have is n -1 .

Test yourself:

1.) Without graphing name the possible numbers of extrema for:

2.) Fill in the blanks using the same functions as in number 1.

Use the following pictures to answer questions 3 through 5.

3.) What are the possible degrees of the polynomials in the graphs?

Deg F = ____________________________

Deg G = ____________________________

Deg H = ____________________________

4.) Is the leading coefficient positive or negative for:

F(x)? _____________ G(x)? _____________ H(x)? ___________

5.)

Answers:

1.) f: 6, 4, 2, 0         g: 9, 7, 5, 3, 1        h: 3, 1         k: 4, 2, 0

2.)

3.)    F: 4, 6, 8, 10, 12, 14, 16, 18, ...
        G: 5, 7, 9, 11, 13, 15, 17, 19, ...
        H: 4, 6, 8, 10, 12, 14, 16, 18, ... * *Note: The reason H can't have degree 2, is because there is a little dip that looks as if it were starting to make 2 more extrema, but it doesn't dip low enough to actually add the extrema. It still affects the degree. However, if this were a test question and you wrote 2 as a possibility, I would not count it wrong.

4.) F: positive        G: positive         H: negative

5.)