5.8 Analyze Graphs of Polynomial Functions

 

Local maximum

The y-coordinate of a turning a point if the point is higher than all nearby points

 

Local minimum

The y-coordinate of a turning a point if the point is higher than all nearby points

 

ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTS

 

Let f(x) = an x n + an _ 1xn _ 1 +…+ a1x + a0 be a polynomial function. If k is a real number, than the following statements are equivalent.

Zero: _k_ is a zero of the polynomial function f.

Factor: _x— k_ is a factor of the polynomial f(x).

Solution: _k_ is a solution of the polynomial equation f(x) = 0.

x-intercept: _k_ is an x-intercept of the graph of the polynomial function f. The graph of f contains (_k_, 0).

 

Example 1

Use x-intercepts to graph a polynomial function

 

Graph the function

f(x) =      (x + l)2(x - 4).

 

1.  Use the intercepts. Because _-1_ and _4_ are zeros of f, plot (_-1_, _0_ ) and
(_4_ , _0_ ).

2.    Plot points between and beyond the x-intercepts.

x

-2

0

1

2

3

5

y

-

_-1_

_-3_

-

_-4_

_9_

 

3.      Determine the end behavior. Because f has _three_ factors of the form x - k, and a constant factor of -    it is a _cubic_ function with a _positive_ leading coefficient. So, f(x) ® _-¥_as x ® - ¥ ® as x and f(x) ® + ¥_ as x ® + ¥.

 

 

 

 

 

 

4.      Draw the graph so that it passes through the plotted points and has the appropriate end behavior.

 

 


TURNING POINTS OF POLYNOMIAL FUNCTIONS

 

The graph of every polynomial function of degree n has at most _n - 1_ turning points. Moreover, if a polynomial function has n distinct real zeros, then its graph has exactly
_n - 1_ turning points.

 

Example 2

Find turning points

 

Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur.

a.  f(x) = x3 - 4x2 + 6

b. f(x) = -x4 + 3x3 + x2 - 4x

a. Use a graphing calculator to graph the function.

Notice that the graph of f has _three_ x-intercepts and _two_ turning points. Use the graphing calculator's zero, maximum, and minimum features to approximate the coordinates of the points.

 

 


The x-intercepts of the graph are _x » -1.09, >x » 1.57, and x » 3.51_. The function has a local maximum at (_0_, _6_) and a local minimum at (_2.67_, -3.48_).

 

 

 

 

 

 

b.  Use a graphing calculator to graph the function.

Notice that the graph has _four_ x-intercepts and _three_ turning points. Use the graphing calculator's zero, maximum, and minimum features to approximate the coordinates of the points.

 

The x-intercepts of the graph are _x » -1.11, x= 0, x » 1.25, and x » 2.86_. The function has local maximums at (_-0.68, 2.03_) and (_2.28_, _4.61_ ). The function has a local minimum at (_0.65_, -1.53_).