You can print out these notes to follow along with the video below and keep notes to organize your thoughts.

We will work with two specific rational functions: $f(x) = \frac {1}{x}$ and $g(x)=\frac {1}{x^2}$. By using what we know about shifting and leading coefficients from Quadratics, Radicals, and Polynomials, we have two basic equations for rational functions:

$$f(x) = \frac {a}{x-h} + k$$

$$\graph [rectangular]{f(x) = a/(x-h) + k, a=1, h=0, k=0, x=h, y=k}$$

$$g(x) = \frac {a}{(x-h)^2} + k$$

$$\graph [rectangular]{f(x) = a/(x-h)^2 + k, a=1, h=0, k=0, x=h, y=k}$$

Thinking back to the previous objective, our rational functions are not defined at $x=h$. So while $(h, k)$ acts like our vertex for quadratics, it is not actually a point on the graph! Check out the Desmos graphs to see how $a$, $h$, and $k$ affect the graphs of these two functions.

We will focus on working from graphs to the equation. If you master this, you’ll be able to work backwards and graph a radical function from the equation.