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Section 4.2
Graphing Polynomial Functions
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
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Objectives
Graph polynomial functions. Use the intermediate value theorem to determine whether a function has a real zero between two given real numbers.
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Graphing Polynomial Functions
If P(x) is a polynomial function of degree n, the graph of the function has:
at most n real zeros, and thus at most n x- intercepts; at most n 1 turning points.
(Turning points on a graph, also called relative maxima and minima, occur when the function changes from decreasing to increasing or from increasing to decreasing.)
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To Graph a Polynomial Function 1. Use the leading-term test to determine the end behavior.2. Find the zeros of the function by solving f (x) = 0. Any real
zeros are the first coordinates of the x-intercepts.3. Use the x-intercepts (zeros) to divide the x-axis into intervals
and choose a test point in each interval to determine the sign of all function values in that interval.
4. Find f (0). This gives the y-intercept of the function.5. If necessary, find additional function values to determine the
general shape of the graph and then draw the graph.6. As a partial check, use the facts that the graph has at most n
x-intercepts and at most n 1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x-axis. We can also check the graph with a graphing calculator.
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Example
Graph the polynomial function f (x) = 2x3 + x2 8x 4.
Solution:1. The leading term is 2x3. The degree, 3, is odd, the coefficient, 2, is positive. Thus the end behavior of the graph will appear as:
2. To find the zero, we solve f (x) = 0. Here we can use factoring by grouping.
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Example continued
Factor:
The zeros are 1/2, 2, and 2. The x-intercepts are (2, 0), (1/2, 0), and (2, 0).
3. The zeros divide the x-axis into four intervals:(, 2), (2, 1/2), (1/2, 2), and (2, ).We choose a test value for x from each interval and find f(x).
3 2
2
2
2 8 4 0
(2 1) 4(2 1) 0
(2 1)( 4) 0(2 1)( 2)( 2) 0
x x x
x x x
x xx x x
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Example continued
4. To determine the y-intercept, we find f(0):
The y-intercept is (0, 4).
3 2
3 2
( ) 2 8 4
( ) 2( ) 8( )0 0 0 0 4 4
f x x x x
f
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Example continued
5. We find a few additional points and complete the graph.
6. The degree of f is 3. The graph of f can have at most 3 x-intercepts and at most 2 turning points. It has 3 x-intercepts and 2 turning points. Each zero has a multiplicity of 1; thus the graph crosses the x-axis at 2, 1/2, and 2. The graph has the end behavior described in step (1). The graph appears to be correct.
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Intermediate Value Theorem
For any polynomial function P(x) with real coefficients, suppose that for a b, P(a) and P(b) are of opposite signs. Then the function has a real zero between a and b.
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Example
Using the intermediate value theorem, determine, if possible, whether the function has a real zero between a and b. a) f(x) = x3 + x2 8x; a = 4 b = 1 b) f(x) = x3 + x2 8x; a = 1 b = 3
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Solution
We find f(a) and f(b) and determine where they differ in sign. The graph of f(x) provides a visual check.
f(4) = (4)3 + (4)2 8(4) = 16
f(1) = (1)3 + (1)2 8(1) = 8
By the intermediate value theorem, since f(4) and f(1) have opposite signs, then f(x) has a zero between 4 and 1.
zero
y = x3 + x2 8x
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Solution
f(1) = (1)3 + (1)2 8(1) = 6
f(3) = (3)3 + (3)2 8(3) = 12
By the intermediate value theorem, since f(1) and f(3) have opposite signs, then f(x) has a zero between 1 and 3.
zero
y = x3 + x2 8x