# Lesson 19

End Behavior of Rational Functions

• Let’s explore the end behavior of rational functions.

### Problem 1

The function $$f(x)=\frac{5x+2}{x-3}$$ can be rewritten in the form $$f(x)=5+\frac{17}{x-3}$$. What is the end behavior of $$y=f(x)$$?

### Problem 2

Rewrite the rational function $$g(x) = \frac{x^2+7x-12}{x+2}$$ in the form $$g(x) = p(x)+ \frac{r}{x+2}$$, where $$p(x)$$ is a polynomial and $$r$$ is an integer.

### Problem 3

Match each polynomial with its end behavior as $$x$$ gets larger and larger in the positive and negative directions. (Note: Some of the answer choices are not used and some answer choices are used more than once.)

### Problem 4

Let the function $$P$$ be defined by $$P(x) = x^3 + 2x^2 - 13x + 10$$. Mai divides $$P(x)$$ by $$x+5$$ and gets:

$$\displaystyle \require{enclose} \begin{array}{r} x^2-3x+2 \phantom{00}\\ x+5 \enclose{longdiv}{x^3+2x^2-13x+10} \\ \underline{-x^3-5x^2} \phantom{-13x+100} \\ -3x^2-13x \phantom{+100}\\ \underline{3x^2+15x} \phantom{+100} \\ 2x+10 \\ \underline{-2x-10} \\ 0 \end{array}$$

How could we tell by looking at the remainder that $$(x+5)$$ is a factor?

(From Unit 2, Lesson 13.)

### Problem 5

For the polynomial function $$f(x)=x^4+3x^3-x^2-3x$$ we have $$f(\text-3)= 0, f(\text-2)=\text-6, f(\text-1)=0$$, $$f(0)=0, f(1)=0,f(2)=30, f(3)=144$$. Rewrite $$f(x)$$ as a product of linear factors.

(From Unit 2, Lesson 15.)

### Problem 6

There are many cones with a volume of $$60\pi$$ cubic inches. The height $$h(r)$$ in inches of one of these cones is a function of its radius $$r$$ in inches where $$h(r)=\frac{180}{r^2}$$.

1. What is the height of one of these cones if its radius is 2 inches?
2. What is the height of one of these cones if its radius is 3 inches?
3. What is the height of one of these cones if its radius is 6 inches?
(From Unit 2, Lesson 16.)

### Problem 7

A cylindrical can needs to have a volume of 10 cubic inches. There needs to be a label around the side of the can. The function $$S(r)=\frac{20}{r}$$ gives the area of the label in square inches where $$r$$ is the radius of the can in inches.

1. As $$r$$ gets closer and closer to 0, what does the behavior of the function tell you about the situation?
2. As $$r$$ gets larger and larger, what does the end behavior of the function tell you about the situation?
(From Unit 2, Lesson 17.)

### Problem 8

Match each rational function with a description of its end behavior as $$x$$ gets larger and larger.

(From Unit 2, Lesson 18.)