171S4.2q Graphing Polynomial Functions
March 22, 2013
4.2 Graphing Polynomial Functions
MAT 171 Precalculus Algebra
Dr. Claude Moore
Cape Fear Community College
CHAPTER 4: Polynomial and Rational Functions
4.1 Polynomial Functions and Models
4.2 Graphing Polynomial Functions
4.3 Polynomial Division; The Remainder and Factor Theorems
4.4 Theorems about Zeros of Polynomial Functions
4.5 Rational Functions
4.6 Polynomial and Rational Inequalities
• Graph polynomial functions.
• Use the intermediate value theorem to determine whether a function has a real zero between two given real numbers.
Click the globe to the left and visit SAS Curriculum Pathways for interactive programs on Polynomial Functions. User: able7oxygen
Quick Launch: 1022 (Polynomial patterns ­ worksheet), 1441 (Exploring Graphs of Polynomial Functions)
Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
You may use the "Polynomial Roots" program to graph polynomial functions and find the real roots (zeros). http://cfcc.edu/mathlab/geogebra/poly_roots.html
The following PowerPoint presentation was developed by Dr. Moore. It covers material in sections as indicated below.
Section 4.1, Slides 7­12 http://cfcc.edu/faculty/cmoore/c2s1­4.ppt
Section 4.2, Slides 13­14 http://cfcc.edu/faculty/cmoore/c2s1­4.ppt
Section 4.3, Slides 15­20 http://cfcc.edu/faculty/cmoore/c2s1­4.ppt
Section 4.4, Slides 21­26 http://cfcc.edu/faculty/cmoore/c2s1­4.ppt
Oct 18­1:28 PM
Oct 18­1:28 PM
Example
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 x­intercepts;
at most n ­ 1 turning points.
(Relative maxima and minima occur when the function changes from decreasing to increasing or from increasing to decreasing.)
Steps to Graph a Polynomial Function 1. Use the leading­term 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 x­intercepts.
3. Use the x­intercepts (zeros) to divide the x­axis 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 y­intercept 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 x­intercepts 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 x­axis. We can also check the graph with a graphing calculator.
Oct 18­1:28 PM
Example continued
­1 ­1/2
2. To find the zero, we solve f (x) = 0. Here we can use factoring by grouping. Factor:
The zeros are ­1/2, ­2, and 2. The x­intercepts are (­2, 0), (­1/2, 0), and (2, 0).
3. The zeros divide the x­axis 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).
Intermediate Value Theorem
For any polynomial function P(x) with real coefficients, (­2, 0), (­1/2, 0), and (2, 0).
­2
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:
Oct 18­1:28 PM
The zeros are ­1/2, ­2, and 2. The x­intercepts are ­3
Graph the polynomial function f (x) = 2x3 + x2 ­ 8x ­ 4.
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.
1
2
4. To determine the y­intercept, we find f(0):
f(x) = 2x3 + x2 ­ 8x ­ 4
f(0) = 2(0)3 + (0)2 ­ 8(0) ­ 4 = ­4.
The y­intercept is (0, ­4).
5. We find a few additional points and complete the graph.
3
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
Mathematica Interactive Figures are available through Tools for Success, Activities and Projects in CourseCompass. You may access these through CourseCompass or from the Important Links webpage. You must Login to MML to use this link.
6. The degree of f is 3. The graph of f can have at most 3 x­intercepts and at most 2 turning points. It has 3 x­intercepts and 2 turning points. Each zero has a multiplicity of 1; thus the graph crosses the x­axis at ­2, ­1/2, and 2. The graph has the end behavior described in step (1). The graph appears to be correct.
Oct 18­1:28 PM
Oct 18­1:28 PM
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171S4.2q Graphing Polynomial Functions
March 22, 2013
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)3 ­ 8(­1) = 8
323/2. For the function f(x) = ­x2 + x4 ­ x6 + 3, state: a) the maximum number of real zeros that the function can have; b) the maximum number of x­intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have.
By the intermediate value theorem, since f(­4) and f(­1) have opposite signs, then f(x) has a zero between ­4 and ­1.
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.
Oct 18­1:28 PM
Oct 20­9:44 AM
323/4. For the function f(x) = (1/4)x3 + 2x2 , state: a) the maximum number of real zeros that the function can have; b) the maximum number of x­intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have.
323/5. For the function f(x) = ­x ­ x3 , state: a) the maximum number of real zeros that the function can have; b) the maximum number of x­intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have.
Oct 20­9:44 AM
Oct 20­9:44 AM
323/6. For the function f(x) = ­3x4 + 2x3 ­ x ­ 4, state: a) the maximum number of real zeros that the function can have; b) the maximum number of x­intercepts that the graph of the function can have; and c) the maximum number of turning points that the graph of the function can have.
323/8. Use the leading­term test and your knowledge of y­intercepts to match the function, f(x) = ­0.5x6 ­ x5 + 4x4 ­ 5x3 ­ 7x2 + x ­ 3, with one of the graphs (a) ­ (f) below.
323/10. Use the leading­term test and your knowledge of y­intercepts to match the function, f(x) = (­1/3)x3 ­ 4x2 + 6x + 42, with one of the graphs (a) ­ ( f ) above.
Oct 20­9:44 AM
Oct 20­9:44 AM
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171S4.2q Graphing Polynomial Functions
323/11. Use the leading­term test and your knowledge of y­intercepts to match the function, f(x) = x4 ­ 2x3 + 12x2 + x ­ 20, with one of the graphs (a) ­ (f) below.
March 22, 2013
323/14. Graph the polynomial function. Follow the steps outlined in the procedure on p. 317: g(x) = x4 ­ 4x3 + 3x2 .
1. Use the leading­ term test to determine the end behavior. Y
2. Find the zeros of the function by solving . 3. Use the x­ intercepts ( zeros). 4. Find f(0). X
5. If necessary, find additional function values.
6. The graph has at most n x­ intercepts and at most turning points. 7. Multiplicity of zeros: graph crosses or is tangent to the x­axis.
323/12. Use the leading­term test and your knowledge of y­intercepts to match the function, f(x) = ­0.3x7 + 0.11x6 ­ 0.25x5 + x4 + x3 ­ 6x ­ 5, with one of the graphs (a) ­ ( f ) above.
The function above was graphed with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­9:44 AM
Oct 20­12:18 PM
323/22. Graph the polynomial function. Follow the steps outlined in the procedure on p. 317: f(x) = (1/2)x3 +(5/2)x2.
1. Use the leading­ term test to determine the end behavior. Y
323/32. Graph the polynomial function. Follow the steps outlined in the procedure on p. 317: h(x) = (x + 2)3 .
1. Use the leading­ term test to determine the end behavior. Y
2. Find the zeros of the function by solving . 2. Find the zeros of the function by solving . 3. Use the x­ intercepts ( zeros). 3. Use the x­ intercepts ( zeros). 4. Find f(0). 4. Find f(0). 5. If necessary, find additional function values.
X
5. If necessary, find additional function values.
6. The graph has at most n x­ intercepts and at most turning points. X
6. The graph has at most n x­ intercepts and at most turning points. 7. Multiplicity of zeros: graph crosses or is tangent to the x­axis.
7. Multiplicity of zeros: graph crosses or is tangent to the x­axis.
The function above was graphed (on the right) with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
The function above was graphed with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
324/36. Graph the polynomial function. Follow the steps outlined in the procedure on p. 317: h(x) = x5 ­ 5x3 + 4x.
1. Use the leading­ term test to determine the end behavior. 324/40. Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b.
f(x) = x3 + 3x2 ­ 9x ­ 13; a = 1, b = 2.
2. Find the zeros of the function by solving . 3. Use the x­ intercepts ( zeros). 4. Find f(0). 5. If necessary, find additional function values.
Y
6. The graph has at most n x­ intercepts and at most turning points. 7. Multiplicity of zeros: graph crosses or is tangent to the x­axis.
X
The graph is below the x­axis between x = 1 and x = 2.
The function above was graphed with the
GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
The function above was graphed with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
Oct 20­12:17 PM
3
171S4.2q Graphing Polynomial Functions
324/42. Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b.
f(x) = 3x2 ­ 2x ­ 11; a = 2, b = 3.
March 22, 2013
322/44. Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b. f(x) = 2x5 ­ 7x + 1; a = 1, b = 2.
The graph crosses the x­axis between x = 2 and x = 3.
f(1) = ­4 and f(2) = 51
Since the signs are different, there is a root between x = 1 and x = 2.
The function above was graphed with the
GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
The function above was graphed with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
322/46. Using the intermediate value theorem, determine, if possible, whether the function f has a real zero between a and b. f(x) = x4 ­ 3x2 + x ­ 1; a = ­3, b = ­2.
f(­3) = 50 and f(­2) = 1
Since the signs are the same, we don't know whether there is a root between x = ­3 and x = ­2.
The graph to the left shows that there is a root between x = ­1 and x = ­2.
The function above was graphed with the GeoGebra program available at http://cfcc.edu/mathlab/geogebra/poly_roots.html
Oct 20­12:17 PM
4