Oblique asymptotes

In addition to horizontal and vertical asymptotes, we can also have oblique asymptotes that are slanted lines with equation $y=mx+c$ .

Curves of the form $\displaystyle y = \frac{ax^2+bx+c}{dx+e}$ can be written as $\displaystyle y = Ax+B + \frac{C}{dx+e}$ .
They have a vertical asymptote $x= \displaystyle -\frac{e}{d}$ and a oblique asymptote $y = Ax+B$ .

Examples

{y = \frac{4 x^2 - 5 x - 25}{x - 3}} {= 4 x + 7} - \frac{4}{x - 3}

Asymptotes:

{x=3,}

{y = 4 x + 7.}

Other points of note

In addition to the shape of the curve, asymptotes and axial intercepts, we are also interested in the stationary points of the curve (if any).

Concept check 1: do you know how to use a graphing calculator to get the coordinates of stationary points?

Concept check 2: do you recall how to get the coordinates of stationary points by differentiation?