High School Math : Finding Asymptotes

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #1 : Finding Asymptotes

A function is defined by the following rational equation:

\(\displaystyle f(x)=\frac{x+2}{2x+5}\)

What are the horizontal and vertical asymptotes of this function's graph?

Possible Answers:

\(\displaystyle y=0, x=0\)

\(\displaystyle y=\frac{1}{2}, x=-\frac{5}{2}\)

\(\displaystyle y=-\frac{5}{2}, x=\frac{1}{2}\)

\(\displaystyle y=-2, x=\frac{2}{5}\)

\(\displaystyle y=2, x=0\)

Correct answer:

\(\displaystyle y=\frac{1}{2}, x=-\frac{5}{2}\)

Explanation:

To find the horizontal asymptote, compare the degrees of the top and bottom polynomials. In this case, the two degrees are the same (1), which means that the equation of the horizontal asymptote is equal to the ratio of the leading coefficients (top : bottom). Since the numerator's leading coefficient is 1, and the denominator's leading coefficient is 2, the equation of the horizontal asymptote is \(\displaystyle y=\frac{1}{2}\).

 

To find the vertical asymptote, set the denominator equal to zero to find when the entire function is undefined:

\(\displaystyle 2x+5=0\)

\(\displaystyle 2x=-5\)

\(\displaystyle x=-\frac{5}{2}\)

Example Question #1 : Finding Asymptotes

A function is defined by the following rational equation:

\(\displaystyle f(x)=\frac{x^{2}+x-2}{x+3}\)

What line does \(\displaystyle f(x)\) approach as \(\displaystyle x\) approaches infinity?

Possible Answers:

\(\displaystyle y=x+4\)

\(\displaystyle y=x\)

\(\displaystyle y=x^{2}-2x+4\)

\(\displaystyle y=x+2\)

\(\displaystyle y=x-2\)

Correct answer:

\(\displaystyle y=x-2\)

Explanation:

This question is asking for the equation's slant asymptote. To find the slant asymptote, divide the numerator by the denominator. Long division gives us the following:

\(\displaystyle x-2+\frac{4}{x+3}\)

However, because we are considering \(\displaystyle x\) as it approaches infinity, the effect that the last term has on the overall linear equation quickly becomes negligible (tends to zero). Thus, it can be ignored.

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