Name:
Date: __________________
Algebra 2
Polynomial Graphs Using Multiplicity
Polynomial Graphs Using Multiplicity
There are two possible things a polynomial graph can do at a point on the x-axis. The
graph might pass through the x-axis, or it might just touch the x-axis without passing through.
How do we tell which of these things will happen? It depends on the multiplicity of the factor
associated with that x-intercept (meaning: how many times that factor is repeated).
Recap: End Behavior
Here’s a review of what we’ve learned about end behavior. For a polynomial in factored form
with linear factors, the degree can be found by counting all the factors containing x. If there’s a
numerical factor, that number is the leading coefficient. (If you don’t see a numerical factor, then
the leading coefficient is just 1.) Here are the end behavior rules from the last two lessons,
summarized in a form that could help you remember them:


The leading coefficient tells the end behavior on the right: up if positive, down if negative.
For even degree, the left and right ends go in the same direction. If odd degree, opposite direction.
Example: f(x) = 2 x (x + 3) (x – 4)
The leading coefficient is 2, so the right end of the graph goes up.
The degree is 3 (from counting all factors except the 2) so the ends go in opposite directions;
thus the left end goes down.
The zeros come from the factors: x = 0, x = –3, and x = 4. So these numbers are the x-intercepts.
Here’s a calculator graph of f(x) after choosing an appropriate window. You can see that it has all
of the features listed above.
What About Multiplicity?
If a factored form has a repeated factor, the number of times the factor is repeated is called the
multiplicity of the factor. Very often, the repeated factors are written using powers, in which
case, the multiplicity is just the exponent.
For example, for the function written f(x) = (x – 2)(x – 4)(x – 4)(x – 4) or f(x)=(x – 2)(x – 4)3,
the multiplicity of the (x – 4) factor is 3.
Name:
Date: __________________
Algebra 2
Polynomial Graphs Using Multiplicity
Conclusions on Multiplicity
Suppose that a function in factored form includes (x – c)k, a factor with multiplicity k.
 If the multiplicity k is odd, then the graph passes through the x-axis at x = c.
 If the multiplicity k is even, then the graph touches the x-axis without passing through at x = c.
Example: f(x) = –5 (x – 2)2(x – 4)3(x – 6)
At x = 2, the multiplicity is even (2) so the graph touches the x-axis without passing through.
At x = 4, the multiplicity is odd (3) so the graph passes through the x-axis.
At x = 6, the multiplicity is odd (1) so the graph passes through the x-axis.
The degree is even (6) and the leading coefficient is negative (–5) so the end behavior is down on
both sides.
Here’s a calculator graph of f(x) after choosing an appropriate window. You can see that its
appearance matches what’s stated above.
Problems
1. Let f(x) = 3 (x + 2)3 (x – 1)2 (2x + 9). Do not use your calculator until part d.
a. List the factors and the x-intercepts (zeros) of f(x). In the right column, circle either touch
or pass-through describing what the graph will do at each x-intercept.
factor
x-intercept (zero)
(x + 2)3
appearance at x-intercept
touch pass-through
touch pass-through
touch pass-through
b. What are the degree and the leading coefficient?
degree: _____
leading coefficient: _____
c. What are the end behaviors (up or down)?
left end: _____
right end: _____
d. Now graph f(x) on your calculator. Adjust the window until you can see all of the features
listed above. (You’ll probably need to make Ymin and Ymax larger, without changing
the Xmin and Xmax.) Then, sketch the screen:
\
Name:
Date: __________________
2. Let f(x) = – 101 (x +
1
2
Algebra 2
Polynomial Graphs Using Multiplicity
)2 (x – 3) (x + 2)4. Do not use your calculator until part d.
a. List the factors and the x-intercepts (zeros) of f(x). In the right column, circle either touch
or pass-through describing what the graph will do at each x-intercept.
factor
x-intercept (zero)
appearance at x-intercept
touch pass-through
touch pass-through
touch pass-through
b. What are the degree and the leading coefficient?
degree: _____
leading coefficient: _____
c. What are the end behaviors (up or down)?
left end: _____
right end: _____
d. Now graph f(x) on your calculator. Adjust the window until you can see all of the features
listed above. (You’ll probably need to make Ymin and Ymax larger, without changing
the Xmin and Xmax.) Then, sketch the screen:
3. Let f(x) = – (x + 3) x3 (3x – 2)2. Do not use your calculator until part c.
a. List the factors and the x-intercepts (zeros) of f(x). In the right column, circle either touch
or pass-through describing what the graph will do at each x-intercept.
Factor
x-intercept (zero)
appearance at x-intercept
touch pass-through
touch pass-through
touch pass-through
b. Using the degree and leading coefficient, what are the end behaviors (up or down)?
left end: _____
right end: _____
c. Now graph f(x) on your calculator. Adjust the window until you can see all of the features
listed above. (You’ll probably need to make Ymin and Ymax larger, without changing
the Xmin and Xmax.) Then, sketch the screen:
Name:
Date: __________________
Algebra 2
Polynomial Graphs Using Multiplicity
4. Let f(x) = – 12 (x + 2)2 (x – 1)3. Do not use your calculator until part e.
a. List the x-intercepts of f(x) and describe the graph at each (touch or pass-through).
b. Find the y-intercept of f(x).
Hint: y-intercept is the point where x = 0.
c. Based on the degree and leading coefficient, describe the end behavior for f(x).
d. Draw a possible graph for f(x) having all the features listed above.
Put number labels on the x- and y-intercepts.
e. Now graph f(x) on your calculator. After adjusting the window to get a good view, draw:
f. How does the calculator graph compare to what you drew by yourself? Describe the ways
(if any) that the actual graph (part e) is different from what you originally drew (part d).
Name:
Date: __________________
Algebra 2
Polynomial Graphs Using Multiplicity
5. Let f(x) = x (x + 2) (3x – 2) (x – 4)2 . Do not use your calculator until part e.
a. List the x-intercepts of f(x) and describe the graph at each (touch or pass-through).
b. Find the y-intercept of f(x).
Hint: y-intercept is the point where x = 0.
c. Based on the degree and leading coefficient, describe the end behavior for f(x).
d. Draw a possible graph for f(x) having all the features listed above.
Put number labels on the x- and y-intercepts.
e. Now graph f(x) on your calculator. After adjusting the window to get a good view, draw:
f. How does the calculator graph compare to what you drew by yourself? Describe the ways
(if any) that the actual graph (part e) is different from what you originally drew (part d).
Name:
Date: __________________
Algebra 2
Polynomial Graphs Using Multiplicity
6. Let f(x) = –0.2 (x + 4)2(x – 1)(x + 1)2(x –4). Do not use your calculator until part e.
a. List the x-intercepts of f(x) and describe the graph at each (touch or pass-through).
b. Find the y-intercept of f(x).
Hint: y-intercept is the point where x = 0.
c. Based on the degree and leading coefficient, describe the end behavior for f(x).
d. Draw a possible graph for f(x) having all the features listed above.
Put number labels on the x- and y-intercepts.
e. Now graph f(x) on your calculator. After adjusting the window to get a good view, draw:
f. How does the calculator graph compare to what you drew by yourself? Describe the ways
(if any) that the actual graph (part e) is different from what you originally drew (part d).