Chapter 8.3, Problem 6E

Solution

To identify: The three rational functions that have an oblique asymptote of $y=2x$ . Graph to check your work.

The three rational functions are $y=\frac{2x^2+1}{x},y=\frac{2x^3+x+1}{x^2+1},y=\frac{2x^3}{x^2}$ .

Given information:

The given asymptote is $y=2x$ .

Explanation:

Consider the function shown below:

  $y=\frac{2x^2+1}{x},y=\frac{2x^3+x+1}{x^2+1},y=\frac{2x^3}{x^2}$

These three functions have an oblique asymptote at $y=2x$ because all the functions give the same quotient $2x$ .

Now draw the graph as shown:

Graph of $y=\frac{2x^2+1}{x}$ is in red, the graph of $y=\frac{2x^3+x+1}{x^2+1}$ is in green, and the graph of $y=\frac{2x^3}{x^2}$ is in blue.