# Lesson 19

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?

### Problem 5

For the polynomial function \(f(x)=x^4+3x^3-x^2-3x\) we have \(f(\text-3)= \text-40, f(\text-2)=\text-6, f(\text-1)=0\), \(f(0)=0, f(1)=30,f(2)=30, f(3)=144\). Rewrite \(f(x)\) as a product of linear factors.

### 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 \(h(r)=\frac{180}{r^2}\).

- What is the height of one of these cones if its radius is 2 inches?
- What is the height of one of these cones if its radius is 3 inches?
- What is the height of one of these cones if its radius is 6 inches?

### 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.

- As \(r\) gets closer and closer to 0, what does the behavior of the function tell you about the situation?
- As \(r\) gets larger and larger, what does the end behavior of the function tell you about the situation?

### Problem 8

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