Lesson 18

Graphs of Rational Functions (Part 2)

  • Let’s learn about horizontal asymptotes.

Problem 1

Rewrite the rational function \(g(x) = \frac{x-4}{x}\) in the form \(g(x) = c + \frac{r}{x}\), where \(c\) and \(r\) are constants.

Problem 2

The average cost (in dollars) per mile for riding \(x\) miles in a cab is \(c(x)=\frac{2.5+2x}{x}\). As \(x\) gets larger and larger, what does the end behavior of the function tell you about the situation?

Problem 3

The graphs of two rational functions \(f\) and \(g\) are shown. One of them is given by the expression \(\frac{2-3x}{x}\). Which graph is it? Explain how you know.

graph of y = f of x. horizontal asymptote at y = -2 point 5. x intercept at 0 point 5. 
graph of y = g of x. horizontal asymptote at y = 2 point 5. x intercept at - 0 point 5. 

Problem 4

Which polynomial function’s graph is shown here?

polynomial function with roots of -1, 2, and 5
A:

\(f(x)=(x+1)(x+2)(x+5)\)

B:

\(f(x)=(x+1)(x-2)(x-5)\)

C:

\(f(x)=(x-1)(x+2)(x+5)\)

D:

\(f(x)=(x-1)(x-2)(x-5)\)

(From Unit 2, Lesson 7.)

Problem 5

State the degree and end behavior of \(f(x)=5x^3-2x^4-6x^2-3x+7\). Explain or show your reasoning.

(From Unit 2, Lesson 9.)

Problem 6

The graphs of two rational functions \(f\) and \(g\) are shown. Which function must be given by the expression of \(\frac{10}{x-3}\)? Explain how you know.

graph of y = f of x. vertical asymptote at x = 3. y intercept at -2 point 5. 
graph of y = g of x. vertical asymptote at x = -3. y intercept at 2 point 5. 

(From Unit 2, Lesson 17.)