# Graphing Inequalities Joined by and / Or

Name:Algebra 2

Date:

## 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. Ifthere’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 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.

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.

#### 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 passthrough describing what the graph will do at each x-intercept.

factor / x-intercept (zero) / appearance at x-intercept
(x + 2)3 / touch pass-through
touch pass-through
touch pass-through

b.What are the degree and the 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:

\

2.Let f(x) = –(x + )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 passthrough 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?

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 passthrough 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:

4.Let f(x) = –(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).

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).

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).