Graph z=xy

Rewrite the equation as ${\displaystyle xy=z}$.

Divide each term in ${\displaystyle xy=z}$ by ${\displaystyle x}$ and simplify.

Find where the expression ${\displaystyle \frac{z}{x}}$ is undefined.

The vertical asymptotes occur at areas of infinite discontinuity.

No Vertical Asymptotes

Consider the rational function ${\displaystyle R\left(x\right)=\frac{a{x}^n}{b{x}^m}}$ where ${\displaystyle n}$ is the degree of the numerator and ${\displaystyle m}$ is the degree of the denominator.

1. If ${\displaystyle n<m}$, then the x-axis, ${\displaystyle y=0}$, is the horizontal asymptote.

2. If ${\displaystyle n=m}$, then the horizontal asymptote is the line ${\displaystyle y=\frac{a}{b}}$.

3. If ${\displaystyle n>m}$, then there is no horizontal asymptote (there is an oblique asymptote).

Find ${\displaystyle n}$ and ${\displaystyle m}$.

Since ${\displaystyle n<m}$, the x-axis, ${\displaystyle y=0}$, is the horizontal asymptote.

There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.

No Oblique Asymptotes

This is the set of all asymptotes.

No Vertical Asymptotes

Horizontal Asymptotes: ${\displaystyle y=0}$

No Oblique Asymptotes