Analyzing Graphs of Polynomial Functions

1. Section 4-5
Analyzing Graphs of Polynomial
Functions

2. Essential Questions
• How do you graph polynomial functions and
locate their zeros?
• How do you find the relative maxima and
minima of polynomial functions?

3. Vocabulary
1. Location Principle:

4. Vocabulary
1. Location Principle: Helps us know where to
locate zeros when graphing a polynomial by
examining where f(x) changes signs from
one input of x to another

5. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1

6. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y

7. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3

8. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73

9. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2

10. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9

11. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1

12. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1

13. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0

14. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1

15. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1

16. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3

17. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2

18. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7

19. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3

20. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19

21. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
The real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

22. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

23. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

24. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

25. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

26. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

27. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
x
yThe real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

28. Example 1
Determine consecutive integer values of x between
which each real zero of the function is located. Then
draw the graph.
f (x ) = x 4
− x 3
− 4x 2
+1x y
−3 73
−2 9
−1 −1
0 1
1 −3
2 −7
3 19
The real zeros are
located between
x = −2 and x = −1,
between x = −1
and x = 0, between
x = 0 and x = 1,
and between x = 2
and x = 3

29. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.

30. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y

31. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3

32. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49

33. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2

34. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15

35. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1

36. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1

37. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0

38. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5

39. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1

40. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3

41. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2

42. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1

43. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3

44. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5

45. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

46. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

47. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

48. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

49. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

50. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
x
y

51. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5

52. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
The relative maximum
appears to be near x = 0.

53. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
The relative maximum
appears to be near x = 0.
The relative minimum
appears to be near x = 2.

54. Example 2
f (x ) = x 3
− 3x 2
+ 5
Graph the polynomial. Estimate the x-
coordinates at which the relative maxima and
relative minima occur, then check your
estimation with a graphing calculator.x y
−3 −49
−2 −15
−1 1
0 5
1 3
2 1
3 5
The relative maximum
appears to be near x = 0.
The relative minimum
appears to be near x = 2.
The graphing calculator
confirms these.

55. Example 3
f (x ) = 0.1n3
− 0.6n2
+110
The weight w in pounds of a patient during a 7-
week illness is modeled by the cubic equation
below, where n is the number of weeks since
the patient became ill.
a. Graph the equation.

56. Example 3
f (x ) = 0.1n3
− 0.6n2
+110
The weight w in pounds of a patient during a 7-
week illness is modeled by the cubic equation
below, where n is the number of weeks since
the patient became ill.
a. Graph the equation.
Weeks n
Weightw

57. Example 3
b. Describe the turning
points of the graph and its
end behavior.
Weeks n
Weightw

58. Example 3
b. Describe the turning
points of the graph and its
end behavior.
Weeks n
Weightw
There is a relative
minimum near the 4th
week. The end behavior
as n increases has w also
increasing.

59. Example 3
c. What trends in the
patient’s weight does the
graph suggest?
Weeks n
Weightw

60. Example 3
c. What trends in the
patient’s weight does the
graph suggest?
Weeks n
Weightw
The patient lost weight
during the first four
weeks of the illness, but
put weight back on after
that.

61. Example 3
d. Is it reasonable to
assume the trend will
continue indefinitely?
Weeks n
Weightw

62. Example 3
d. Is it reasonable to
assume the trend will
continue indefinitely?
Weeks n
Weightw
The trend may continue
in the short term, but the
weight of a human
cannot increase
indefinitely.