Algebra 2 / Unit 6: Rational Functions
6.4 Graphs of Rational Functions
Notes
Simplify first. Factors that cancel make holes; factors left in the denominator make vertical asymptotes.
- Vertical asymptote: $x = a$ where the simplified denominator is 0.
- Hole: $x = a$ where a factor cancels from top and bottom.
- Horizontal asymptote (compare degrees of top $n$ and bottom $d$):
- $n < d$: $y = 0$. $n = d$: $\displaystyle y = \frac{\text{leading coeff of top}}{\text{leading coeff of bottom}}$. $n > d$: none.
- Example: $\displaystyle f(x) = \frac{3x}{x - 1}$ has a vertical asymptote $x = 1$ and horizontal asymptote $y = 3$.
Practice
What is the vertical asymptote of $\displaystyle f(x) = \frac{x + 1}{x - 3}$?
The denominator is 0 at $x = 3$, and nothing cancels.
What is the horizontal asymptote of $\displaystyle f(x) = \frac{4x^2 + 1}{2x^2 - 5}$?
Equal degrees: $\displaystyle \frac{4}{2} = 2$.
Describe the graph of $\displaystyle f(x) = \frac{x - 2}{x^2 - 4}$.
$\displaystyle \frac{x - 2}{(x - 2)(x + 2)}$: $x - 2$ cancels (hole), $x + 2$ stays (asymptote).
Ready for more? Take the Unit 6 test.