Rational functions and graphs
Rational functions and graphs
- A rational function is a fraction of two polynomials.
- When the top has higher degree than the bottom, the graph has an oblique (slant) asymptote — find it by dividing out.
- You should relate $y = f(x)$ to $y^2 = f(x)$, $y = \dfrac{1}{f(x)}$, $y = |f(x)|$ and $y = f(|x|)$.
Practice
A rational function has an oblique (slant) asymptote when the degree of the top is:
When the numerator's degree exceeds the denominator's, dividing out leaves a slant line the curve approaches.
Practice
You find an oblique asymptote by:
Polynomial division gives the straight-line part the curve approaches far from the origin.
Practice
A rational function is a fraction of two polynomials.
That is the definition — one polynomial over another.
You've got it
Key idea
- a rational function = polynomial ÷ polynomial
- top degree > bottom degree → an oblique asymptote (found by division)
- learn the related graphs: $y^2 = f$, $1/f$, $|f|$, $f(|x|)$