5.8 Analyze Graphs of Polynomial Functions

Goal Use intercepts to graph polynomial functions.

Your Notes

VOCABULARY

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).

Your Notes

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_
  1. 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 + .
  2. Draw the graph so that it passes through the plotted points and has the appropriate end behavior.

Checkpoint Complete the following exercise.

  1. Graph the function

f(x) = 2 (x 2) (x + 1) (x 1)

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.

Your Notes

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_).

CheckpointComplete the following exercise.

  1. Use a graphing calculator to identify the x-intercepts, local maximums, and local minimums of the graph of f(x) = x4 + x3 5x2 + 4.

x-intercepts: x 2.66, x 0.89;

local maximum: (0, 4)

local minimums: (2, 8) and (1.25, 0.58)

Homework

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