1. LIAL HORNSBY SCHNEIDER
COLLEGE ALGEBRA
LIAL
HORNSBY
SCHNEIDER

2. 3.4 Polynomial Functions: Graphs, Applications, and Models
Graphs of (x) = axn
Graphs of General Polynomial Functions
Turning Points and End Behavior
Graphing Techniques
Intermediate Value and Boundedness Theorems
Approximating Real Zeros
Polynomial Models and Curve Fitting

3. GRAPHING FUNCTIONS OF THE FORM (x) = axn
Example 1
Graph the function. a.
Solution Choose several values for x, and find the corresponding values of (x), or y. x
(x)
– 2
– 8
– 1 1 2 8

4. GRAPHING FUNCTIONS OF THE FORM (x) = axn
Example 1
Graph the function. b.
Solution The graphs of (x) = x3 and g(x) = x5 are both symmetric with respect to the origin. x
g(x)
– 1.5
–7.6
– 1 1
1.5
7.6

5. GRAPHING FUNCTIONS OF THE FORM (x) = axn
Example 1
Graph the function. c.
Solution The graphs of (x) = x3 and g(x) = x5 are both symmetric with respect to the origin. x
(x)
– 2 16
– 1 1 2 x
g(x)
– 1.5
11.4
– 1 1
1.5

6. Graphs of General Polynomial Functions
As with quadratic functions, the absolute value of a in (x) = axn determines the width of the graph. When a> 1, the graph is stretched vertically, making it narrower, while when 0 < a < 1, the graph is shrunk or compressed vertically, so the graph is broader. The graph of (x) = – axn is reflected across the x-axis compared to the graph of (x) = axn.

7. Graphs of General Polynomial Functions
Compared with the graph of the graph of is translated (shifted) k units up if k > 0 andkunits down if k < 0. Also, when compared with the graph of the graph of (x) = a(x – h)n is translated h units to the right if h > 0 and hunits to the left if h < 0.
The graph of shows a combination of these translations. The effects here are the same as those we saw earlier with quadratic functions.

8. EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONS
Example 2
Graph the function. a.
Solution The graph will be the same as that of (x) = x5, but translated 2 units down.

9. In (x) = (x + 1)5, function  has a graph like that of
EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONS
Example 2
Graph the function. b.
Solution
In (x) = (x + 1)5, function  has a graph like that of
(x) = x6, but since
x + 1 = x – (– 1), it is translated 1 unit to the left.

10. Graph the function. c. Solution
EXAMINING VERTICAL AND HORIZONTAL TRANSLATIONS
Example 2
Graph the function. c.
Solution
The negative sign in – 2 causes the graph of the function to be reflected across the x-axis when compared with the graph of (x) = x3. Because  – 2> 1, the graph is stretched vertically as compared to the graph of
(x) = x3. It is also translated 1 unit to the right and 3 units up.

11. Unless otherwise restricted, the domain of a polynomial function is the set of all real numbers. Polynomial functions are smooth, continuous curves on the interval (– , ). The range of a polynomial function of odd degree is also the set of all real numbers. Typical graphs of polynomial functions of odd degree are shown in next slide. These graphs suggest that for every polynomial function  of odd degree there is at least one real value of x that makes (x) = 0 . The real zeros are the x-intercepts of the graph.

12. Odd Degree

13. A polynomial function of even degree has a range of the form (– , k] or [k, ) for some real number k. Here are two typical graphs of polynomial functions of even degree.
Even Degree

14. Recall that a zero c of a polynomial function has as multiplicity the exponent of the factor x – c. Determining the multiplicity of a zero aids in sketching the graph near that zero. If the zero has multiplicity one, the graph crosses the x-axis at the corresponding x-intercept as seen here.

15. If the zero has even multiplicity, the graph is tangent to the x-axis at the corresponding x-intercept (that is, it touches but does not cross the x-axis there).

16. If the zero has odd multiplicity greater than one, the graph crosses the x-axis and is tangent to the x-axis at the corresponding x-intercept. This causes a change in concavity, or shape, at the x-intercept and the graph wiggles there.

17. Turning Points and End Behavior
The previous graphs show that polynomial functions often have turning points where the function changes from increasing to decreasing or from decreasing to increasing.

18. Turning Points
A polynomial function of degree n has at most n – 1 turning points, with at least one turning point between each pair of successive zeros.

19. End Behavior
The end behavior of a polynomial graph is determined by the dominating term, that is, the term of greatest degree. A polynomial of the form
has the same end behavior as

20. End Behavior For instance,
has the same end behavior as It is large and positive for large positive values of x and large and negative for negative values of x with large absolute value.

21. End Behavior
The arrows at the ends of the graph look like those of the graph shown here; the right arrow points up and the left arrow points down.
The graph shows that as x takes on larger and larger positive values, y does also. This is symbolized as
read “as x approaches infinity, y approaches infinity.”

22. End Behavior
For the same graph, as x takes on negative values of larger and larger absolute value, y does also: as

23. End Behavior
For this graph, we have as
and as

24. End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function  of odd degree.
If a > 0, then as and as
Therefore, the end behavior of the graph is of the type that looks like the figure shown here.
We symbolize it as

25. End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function  of odd degree.
2. If a < 0, then as and as
Therefore, the end behavior of the graph looks like the graph shown here.
We symbolize it as

26. End Behavior of Polynomials
Suppose that axn is the dominating term of a polynomial function  of even degree.
If a > 0, then as
Therefore, the end behavior of the
graph looks like the graph shown here.
We symbolize it as

27. End Behavior of Polynomials
Suppose that is the dominating term of a polynomial function  of even degree.
2. If a < 0, then as Therefore, the end behavior of the graph looks like the graph shown here.
We symbolize it as

28. Match each function with its graph.
DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Example 3
Match each function with its graph. A. B. C. D.
Solution Because  is of even degree with positive leading coefficient, its graph is C.

29. Match each function with its graph.
DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Example 3
Match each function with its graph. A. B. C. D.
Solution Because g is of even degree with negative leading coefficient, its graph is A.

30. Match each function with its graph.
DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Example 3
Match each function with its graph. A. B. C. D.
Solution Because function h has odd degree and the dominating term is positive, its graph is in B.

31. Match each function with its graph.
DETERMINING END BEHAVIOR GIVEN THE DEFINING POLYNOMIAL
Example 3
Match each function with its graph. A. B. C. D.
Solution Because function k has odd degree and a negative dominating term, its graph is in D.

32. Graphing Techniques
We have discussed several characteristics of the graphs of polynomial functions that are useful for graphing the function by hand. A comprehensive graph of a polynomial function will show the following characteristics:
1. all x-intercepts (zeros)
2. the y-intercept
3. the sign of (x) within the intervals formed by the x-intercepts, and all turning points
4. enough of the domain to show the end behavior.
In Example 4, we sketch the graph of a polynomial function by hand. While there are several ways to approach this, here are some guidelines.

33. Graphing a Polynomial Function
Let
be a polynomial function of degree n. To sketch its graph, follow these steps.
Step 1 Find the real zeros of . Plot them as x-intercepts.
Step 2 Find (0). Plot this as the y-intercept.

34. Graphing a Polynomial Function
Step 3 Use test points within the intervals formed by the x-intercepts to determine the sign of (x) in the interval. This will determine whether the graph is above or below the x-axis in that interval.

35. Graphing a Polynomial Function
Use end behavior, whether the graph crosses, bounces on, or wiggles through the x-axis at the x-intercepts, and selected points as necessary to complete the graph.

36. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 1 The possible rational zeros are 1, 2, 3, 6, ½, and 3/2. Use synthetic division to show that 1 is a zero.
(1) = 0

37. Graph Solution Step 1 Thus, GRAPHING A POLYNOMIAL FUNCTION Example 4
Factor 2x2 + 7x + 6.
Set each linear factor equal to 0, then solve for x to find real zeros. The three real zeros of  are 1, – 3/2, and – 2.

38. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 2

39. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 3
The x-intercepts divide the x-axis into four intervals: (– , – 2), (– 2, – 3/2), (– 3/2, 1), and (1, ). Because the graph of a polynomial function has no breaks, gaps, or sudden jumps, the values of (x) are either always positive or always negative in any given interval.

40. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 3
To find the sign of (x) in each interval, select an x-value in each interval and substitute it into the equation for (x) to determine whether the values of the function are positive or negative in that interval.

41. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 3
When the values of the function are negative, the graph is below the x-axis, and when (x) has positive values, the graph is above the x-axis.

42. GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 3
A typical selection of test points and the results of the tests are shown in the table on the next slide. (As a bonus, this procedure also locates points that lie on the graph.)

43. Graph Above or Below x-Axis
GRAPHING A POLYNOMIAL FUNCTION
Example 4
Graph
Solution
Step 3
Interval
Test Point
Value of (x)
Sign of (x)
Graph Above or Below x-Axis
(– , – 2)
– 3
– 12
Negative
Below
(– 2, – 3/2)
– 7/4
11/32
Positive
Above
(– 3/2, 1)
– 6
(1, ) 2 28

44. Graph Solution Step 3 GRAPHING A POLYNOMIAL FUNCTION Example 4
Plot the test points and join the x-intercepts, y-intercept, and test points with a smooth curve to get the graph.

45. Graph Solution Step 3 GRAPHING A POLYNOMIAL FUNCTION Example 4
Because each zero has odd multiplicity (1), the graph crosses the x-axis each time. The graph has two turning points, the maximum number for a third-degree polynomial
function.

46. Graph Solution Step 3 GRAPHING A POLYNOMIAL FUNCTION Example 4
The sketch could be improved by plotting the points found in each interval in the table. Notice that the left arrow points down and the right arrow points up. This end behavior is correct since the dominating term of the polynomial is 2x3.

47. Graphing Polynomial Functions
Note If a polynomial function is given in factored form, such as
Step 1 of the guidelines is easier to perform, since real zeros can be determined by inspection. For this function, we see that 1 and 3 are zeros of multiplicity1, and – 2 is a zero of multiplicity 2.

48. Graphing Polynomial Functions
Note Since the dominating term is
the end behavior of the graph is
The y-intercept is

49. Graphing Polynomial Functions
Note
The graph intersects the
x-axis at 1 and 3 but bounces at – 2.
This information is sufficient to quickly sketch the graph of (x).

50. Important Relationships
We emphasize the important relationships among the following concepts.
1. the x-intercepts of the graph of y = (x)
2. the zeros of the function 
3. the solutions of the equation (x) = 0
4. the factors of (x)

51. x-Intercepts, Zeros, Solutions, and Factors
If a is an x-intercept of the graph of
then a is a zero of , a is a
solution of (x) = 0, and x – a is a factor of (x).

52. Intermediate Value Theorem for Polynomials
If (x) defines a polynomial function with only real coefficients, and if for
real numbers a and b, the values (a) and (b) are opposite in sign, then there exists at least one real zero between a and b.

53. LOCATING A ZERO
Example 5
Use synthetic division and a graph to show that (x) = x3 – 2x2 – x + 1 has a real zero between 2 and 3.
Solution Since (2) is negative and (3) is positive, by the intermediate value theorem there must be a real zero between 2 and 3.

54. Caution Be careful how you interpret the intermediate value theorem.
If (a) and (b) are not opposite in sign, it does not necessarily mean that there is no zero between a and b.
In the graph shown here, for example, (a) and (b) are both negative, but – 3 and – 1, which are between a and b, are zeros of (x).

55. Boundedness Theorem
Let (x) be a polynomial function of degree n ≥ 1 with real coefficients and with a positive leading coefficient. If (x) is divided synthetically by x – c, and
(a) if c > 0 and all numbers in the bottom row of the synthetic division are nonnegative, then (x) has no zero greater than c;

56. Boundedness Theorem
(b) if c < 0 and the numbers in the bottom row of the synthetic division alternate in sign (with 0 considered positive or negative, as needed), then (x) has no zero less than c.

57. Proof
We outline the proof of part (a). The proof for part (b) is similar. By the division algorithm, if (x) is divided by x – c , then for some q(x) and r,
where all coefficients of q(x) are nonnegative, r ≥ 0, and c > 0. If x > c, then x – c > 0. Since q(x) > 0 and r ≥ 0,
This means that (x) will never be 0 for x > c.

58. Show that the real zeros of
USING THE BOUNDEDNESS THEOREM
Example 6
Show that the real zeros of
(x) = 2x4 – 5x3 + 3x +1 satisfy the given conditions.
a. No real zero is greater than 3.
Solution Since (x) has real coefficients and the leading coefficient, 2, is positive, use the boundedness theorem. Divide (x) synthetically by x – 3.
All are nonnegative.

59. a. No real zero is greater than 3.
USING THE BOUNDEDNESS THEOREM
Example 6
a. No real zero is greater than 3.
Solution
All are nonnegative.
Since 3 > 0 and all numbers in the last row of the synthetic division are nonnegative, (x) has no real zero greater than 3.

60. b. No real zero is greater than – 1.
USING THE BOUNDEDNESS THEOREM
Example 6
b. No real zero is greater than – 1.
Solution
Divide (x) by x + 1.
These numbers alternate in sign.
Here – 1 < 0 and the numbers in the last row alternate in sign, so (x) has no real zero less than – 1.

61. Approximate the real zeros of (x) = x4 – 6x3 + 8x2 + 2x – 1.
APPROXIMATING REAL ZEROS OF A POLYNOMIAL FUNCTION
Example 7
Approximate the real zeros of
(x) = x4 – 6x3 + 8x2 + 2x – 1.
Solution The greatest degree term is x4, so the graph will have end behavior similar to the graph of (x) = x4, which is positive for all values of x with large absolute values. That is, the end behavior is up at the left and the right,
There are at most four real zeros, since the polynomial is fourth-degree.

62. Approximate the real zeros of (x) = x4 – 6x3 + 8x2 + 2x – 1.
APPROXIMATING REAL ZEROS OF A POLYNOMIAL FUNCTION
Example 7
Approximate the real zeros of
(x) = x4 – 6x3 + 8x2 + 2x – 1.
Solution Since (0) = – 1, the y-intercept is – 1. Because the end behavior is positive
on the left and the right, by the intermediate value theorem  has at least one zero on either side of x = 0. To approximate the zeros, we use a graphing calculator.

63. Approximate the real zeros of (x) = x4 – 6x3 + 8x2 + 2x – 1.
APPROXIMATING REAL ZEROS OF A POLYNOMIAL FUNCTION
Example 7
Approximate the real zeros of
(x) = x4 – 6x3 + 8x2 + 2x – 1.
Solution The graph below shows that there are four real zeros, and the table indicates that they are between – 1 and 0, 0 and 1, 2 and 3, and 3 and 4 because there is a sign change in (x) in each case.

64. APPROXIMATING REAL ZEROS OF A POLYNOMIAL FUNCTION
Example 7
Solution Using the capability of the calculator, we can find the zeros to a great degree of accuracy. The graph shown here shows that the negative zero is approximately –
Similarly, we find that the other three zeros are approximately
, ,
and

65. EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120
The table shows the number of transactions, in millions, by users of bank debit cards for selected years.

66. EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
a. Using x = 0 to represent 1995, x = 3 to represent 1998, and so on, use the regression feature of a calculator to determine the quadratic function that best fits the data. Plot the data and graph.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

67. a. The best-fitting quadratic function for the data is defined by
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120
a. The best-fitting quadratic function for the data is defined by

68. a. The regression coordinates screen is shown below.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
a. The regression coordinates screen is shown below.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

69. a. The graph is shown below.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
a. The graph is shown below.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

70. b. Repeat part (a) for a cubic function.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
b. Repeat part (a) for a cubic function.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

71. b. The best-fitting cubic function is shown below and is defined by
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120
b. The best-fitting cubic function is shown below and is defined by

72. b. The graph of the best-fitting cubic function is shown below.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120
b. The graph of the best-fitting cubic function is shown below.

73. c. Repeat part (a) for a quartic function.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
c. Repeat part (a) for a quartic function.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

74. c. The best-fitting quartic function is defined by
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120
c. The best-fitting quartic function is defined by

75. c. The graph of the best-fitting quartic function is shown below.
EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
Solution
c. The graph of the best-fitting quartic function is shown below.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

76. EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE
Example 8
d. The correlation coefficient, R, is a measure of the strength of the relationship between two variables. The values of R and R2 are used to determine how well a regression model fits a set of data. The closer the value of R2 is to 1, the better the fit. Compare R2 for the three functions to decide which function fits the data.
Year
Transactions
(in millions)
1995
829
1998
3765
2000
6797
2004
14,106
2009
22,120

77. Solution EXAMINING A POLYNOMIAL MODEL FOR DEBIT CARD USE Example 8
d. Find the correlation values R2. See the graph for the quadratic function. The others are for the cubic function and 1 for the quartic function. Therefore, the quartic function provides the best fit.