1. Analyze Graphs of Polynomial Functions
Objectives:
To approximate x-intercepts of a polynomial function with a graphing utility
To locate and use relative extrema of polynomial functions
To sketch the graphs of polynomial functions

2. 5.8: Analyze Graphs of Polynomial Functions
Objective 1
You will be able to approximate x-intercepts of a polynomial function with a graphing utility
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3. Warm-Up
Find the minimum or maximum point for the function f (x) = 3x2 – 6x + 2
The vertex marks the minimum point

4. Warm-Up Find the x-intercepts for the function f (x) = 3x2 – 6x + 2
The x-intercepts or zeros

5. Use the Rational Zero Theorem to find them
Approximating Zeros
We know that the real zeros of a function (x-intercepts) come in two flavors:
Rational:
Use the Rational Zero Theorem to find them
Irrational:
Use the Quadratic Formula to find them after having whittled your function down to a quadratic
Alternatively, we could use the “zero” feature of your graphing calculator.

6. Approximating Zeros

7. Approximating Zeros
*Straight from the manufacturer’s mouth.

8. Exercise 1
Use a graphing utility to find the approximate the real zeros of the function.
f(x) = 4x5 + 12x4 – x – 3

9. Exercise 2
Use a graphing utility to find the approximate the real zeros of the function.
f(x) = 3x5 + 2x4 – 8x3 + 4x2 – x – 1

10. 5.8: Analyze Graphs of Polynomial Functions
Exercise 3
A tachometer measures the speed (in revolutions per minute: RPM) at which an engine shaft rotates.

11. Exercise 3
For a certain boat, the speed x of the engine shaft (in 100s of RPMs) and the speed s of the boat (in miles per hour) are modeled by s(x) = x3 – 0.225x x – 11 What is the tachometer reading when the boat travels 15 mph?

12. 5.8: Analyze Graphs of Polynomial Functions
Objective 2
You will be able to locate and use relative extrema of polynomial functions
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13. Turning Points
The vertex of a parabola marks the turning point of the graph of a quadratic function.
A turning point is a point at which the function values “turn” from increasing to decreasing or vice versa.

14. Turning Points
The y-coordinate of a parabola’s turning point marks the absolute minimum or maximum of the function since there are no other points above or below it.

15. Extrema
Other polynomial functions also have various turning points that mark minimums and maximums; however, they may not be absolute.
Extrema (min/max values) come in two varieties:
Absolute
Relative (Local)

16. Relative Extrema Relative (Local) Maximum:
The y-coordinate of a turning point that is higher on a graph than its surrounding points
Relative (Local) Minimum:
The y-coordinate of a turning point that is lower on a graph than its surrounding points

17. Approximating Extrema
Similar to approximating the location of the zeros of a function, your graphing utility can be used to find extrema. (What follows is a highly detailed set of instructions that will unerringly guide you through this complex process.)

18. Approximating Extrema
It’s a fairly easy exercise to approximate the location of relative extrema using your graphing utility.
Press Y= and enter the function.
Choose your favorite ZOOM setting.
Press 2nd TRACE for the CALC menu.
Choose minimum or maximum.
Set the left bound, right bound, and a guess.
Magic!

19. Exercise 4
Use a graphing utility to find the extrema of the function below.
h(x) = 0.5x3 +x2 – x + 2

20. Exercise 5
Use a graphing utility to find the extrema of the function below.
j(x) = x4 + 3x3 – x2 – 4x – 5

21. Exercise 6
Rooting around in your room while looking for a lost library book, you come across a 24 inch by 24 inch sheet of some sturdy but flexible material of alien origin. How it got there is not important. What is important is that you believe it would be the perfect material to construct an open box with locking tabs to hold all of your toys. You plan to construct this box by cutting congruent squares from each corner and folding along the dotted lines as shown in the diagram.

22. Exercise 6a
Write a function for the volume of your box in terms of x, the side length of the squares you plan to cut from each corner, and then state the domain.
(Of alien origin)

23. Exercise 6b
Use your graphing calculator to find the side length of the square that gives you the most room for all of your toys.
(Of alien origin)

24. 5.8: Analyze Graphs of Polynomial Functions
Objective 3
5.8: Analyze Graphs of Polynomial Functions
You will be able to sketch the graphs of polynomial functions
Six Flags El Toro
Photo by Joel A. Rogers

25. Exercise 7a
For a polynomial function of degree n, how many tuning points will it have?
n = 2

26. Exercise 7b
For a polynomial function of degree n, how many tuning points will it have?
n = 3

27. Exercise 7c
For a polynomial function of degree n, how many tuning points will it have?
n = 4

28. Then f has at most n – 1 turning points
Let f be a polynomial function of degree n.
Until we learn calculus, we have to find relative extrema by plugging in points.
Then f has at most n – 1 turning points
If f has n distinct real roots, then f has exactly n – 1 turning points

29. Interesting Bits Theorem
This is where all the turny bits happen.
If f is a polynomial function of degree n, then its graph only does really interesting things near and between its zeros.

30. Behavior Near Zeros Real Zeros:
Only real zeros are x-intercepts. Imaginary zeros do not touch the x-axis.
Odd Multiplicity:
A zero of odd multiplicity crosses the x-axis at that zero.
Even Multiplicity:
A zero of even multiplicity is tangent to the x-axis at that zero.
Odd
Multiplicity
Even
Multiplicity

31. Graphing Algorithm To graph a polynomial function:
Find and plot the intercepts (x- and y-)
Determine the end behavior
Determine behavior near zeros
Plot points near and between zeros (relative or absolute min/max)
Connect the points with a reasonable curve keeping in mind the end behavior

32. Exercise 8
Graph each of the following polynomial functions the old-fashioned way (in other words, by hand).
𝑓(𝑥)=0.25(𝑥+2)(𝑥−1)(𝑥−3)
𝑔 𝑥 =− 𝑥 𝑥−2 2

33. Exercise 9
Graph each of the following polynomial functions the old-fashioned way (in other words, by hand).
𝑔 𝑥 =2(𝑥−1)2(𝑥−5)
𝑓 𝑥 = 𝑥 𝑥−4

34. Analyze Graphs of Polynomial Functions
Objectives:
To approximate x-intercepts of a polynomial function with a graphing utility
To locate and use relative extrema of polynomial functions
To sketch the graphs of polynomial functions
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