Lesson 2 (Graphing Polynomials)

When you're graphing (or looking at a graph of) polynomials, it can help to already have an idea of what basic polynomial shapes look like. One of the aspects of this is "end behavior", and it's pretty easy. Look at these graphs:


As you can see, even-degree polynomials are either "up" on both ends or "down" on both ends, depending on whether the polynomial has, respectively, a positive or negative leading coefficient. On the other hand, odd-degree polynomials have ends that head off in opposite directions. If they start "down" and go "up", they're positive polynomials; if they start "up" and go "down", they're negative polynomials. All even-degree polynomials behave, on their ends, like quadratics, and all odd-degree polynomials behave, on their ends, like cubics.

The real (that is, non-complex) zeroes of a polynomial correspond to the x-intercepts of the graph of that polynomial. So you can find the number of real zeroes of a polynomial by looking at the graph, and conversely you can tell how many times the graph is going to touch or cross the x-axis by looking at the zeroes of the polynomial (or the factored form of the polynomial). A zero has a "multiplicity", which refers to the number of times that its associated factor appears in the polynomial. For instance, the quadratic (x + 3)(x - 2) has the zeroes x = -3 and x = 2, each occurring once. The eleventh-degree polynomial (x + 3)4(x - 2)7 has the same zeroes, but in this case, x = -3 has multiplicity 4 and x = 2 has multiplicity 7, because of the number of times their factors occur.

The point of multiplicities with respect to graphing is that any factors that occur an even number of times (twice, four times, six times, etc) are squares, so they don't change sign. Squares are always positive. This means that the x-intercept corresponding to an even-multiplicity zero can't cross the x-axis, because the zero can't cause the graph to change sign from positive (above the x-axis) to negative (below the x-axis), or vice versa. The practical upshot is that an even-multiplicity zero makes the graph just barely touch the x-axis, and then turns it back around the way it came. You can see this in the following graphs:


All four graphs have the same zeroes, at x = -6 and at x = 7, but the multiplicity of the zero determines whether the graph crosses at that zero or turns back the way it came.

I was able to compute the multiplicities of the zeroes in part from the fact that the multiplicities will add up to the degree of the polynomial, or two less, or four less, etc, depending on how many complex zeroes there might be. But multiplicity problems don't usually get into complex numbers.

It isn't standard terminology, and you'll learn the proper terms when you get to calculus, but I refer to the "turnings" of a polynomial graph as its "bumps".

For instance, the following graph has three bumps, as indicated by the arrows:

Compare the numbers of bumps in the graphs below to the degrees of their polynomials:


You can see from these graphs that, for degree n, the graph will have, at most, n - 1 bumps. The bumps represent the spots where the graph turns back on itself and heads back the way it came. This change of direction often happens because of the polynomial's zeroes or factors. But extra pairs of factors don't show up in the graph as much more than just a little extra flexing or flattening in the graph.

Because pairs of factors have this habit of disappearing from the graph (or hiding as a little bit of extra flexure or flattening), the graph may have two fewer, or four fewer, or six fewer, etc, bumps than you might otherwise expect, or it may have flex points instead of some of the bumps. That is, the degree of the polynomial gives you the upper limit (the ceiling) on the number of bumps possible for the graph, and the number of bumps gives you the lower limit (the floor) on degree of the polynomial.

To help you keep straight when to add and when to subtract, remember your graphs of quadratics and cubics. Quadratics are degree-two polynomials and have one bump (always); cubics are degree-three polynomials and have two bumps or none (with a flex point instead). So going from your polynomial to your graph, you subtract, and going from your graph to your polynomial, you add. If you know your quadratics and cubics very well, and if you remember that you're dealing with families of polynomials and their family characteristics, you shouldn't have any trouble with this sort of problem.

Sketching Polynomial Graphs

Once you know the basic behavior of polynomial graphs, you can quickly sketch rough graphs, if required. This can save you the trouble of trying to plot a zillion points for a degree-seven polynomial, for instance. Once the graph starts heading off to infinity, you know that the graph is going to keep going, so you can just draw the line heading off the top or bottom of the graph; you don't need to plot a bunch of actual points.




Grading for this Lesson:

To get a 10
: All answers are correct the first time, or within first revision. 
To get a 9: You can have 1 incorrect answer after your original submission.
To get an 8: You can have 2 incorrect answers after your original submission.
To get a 7: You can have 3  incorrect answers after your original submission. 
To get a 6: You can have 4 incorrect answers after your original submission.
To get a 5: Cheating - Plagiarism - purposeful or mistaken, which will lower your final grade for the course (so be very careful when posting your work!); lack of effort, disrespect, or attitude (we are here to communicate with you if you don't understand something); 

Note:  For this class it is necessary to post the questions over each answer. Failure to do so will result in asking for a revision.   No grade will be given for incomplete work.


Assignment

Do the test below. If you get 3 or fewer wrong, it will be assumed that you understand the lesson, and you can go on immediately to the next lesson.

If you get more than 3 wrong, you will be asked to resubmit the wrong answers and show your work. The teacher will then look at your work and give you advice on what you are doing wrong.



Name:
Enter your correct email address:


For questions 1-5, describe the end behavior of the given function.

1.

A. both ends open up
B. both ends open down
C. starts up and ends down
D. starts down and ends up
E. both ends open to the left
F. F. both ends open right

 

2.

A. both ends open down
B. starts down and ends up
C. starts up and ends down
D. both ends open up
E. both ends open to the left
F. both ends open right

 

3.

A. both ends open up
B. starts down and ends up
C. both ends open to the left
D. both ends open right
E. both ends open down
F. starts up and ends down

 

4.

A. starts up and ends down

B. starts down and ends up
C. both ends open up
D. both ends open down
E. both ends open to the left
F. both ends open right

 

5.

A. starts down and ends up
B. both ends open to the left
C. starts up and ends down
D. both ends open up

E. both ends open right
F. both ends open down


For questions 6-10, use the given graph to find all zeros of the function and find the multiplicity of each zero. (Assume all zeros are integers and assume that all zeros have a multiplicity of 1 or 2.)

6.

7.

8.

9.

10.



For questions 11-20, choose the correct graph.

 

11. Which of the following could be a graph of the function?

A.
B.
c.
D.
E.
F.

 

12. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

13. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

14. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

15. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

16. Which of the following could be a graph of the function

A
B

C

D

E

F

 

17. Which of the following could be a graph of the function?

A.
B

C

D

E

F

 

18. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

19. Which of the following could be a graph of the function?

A
B

C

D

E

F

 

20. Which of the following could be a graph of the function?

A
B

C

D

E

F





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