Rational functions
Divide one polynomial by another and the graph breaks apart and runs off along lines. Where are the breaks, which lines does it follow, and what does it mean when it models something real?
Where you are 5 inside
What this area is about
A rational function is a quotient \displaystyle f(x) = \frac{p(x)}{q(x)} of two polynomials, with q not the zero polynomial. Its formula breaks only where q(x) = 0, and near those inputs, or far out to the left and right, its graph often draws closer and closer to a line, an asymptote. This area finds those places from the formula and puts them together into a graph.
How these fit together
Where it lives. Domain and zeros of a rational function removes every input that makes the denominator 0 and finds the inputs whose output is 0.
The lines it follows. Asymptotes is an area of its own: vertical asymptotes at refused inputs that do not cancel away, and horizontal or slant asymptotes far out, all read from the formula.
Procedures. Graphing rational functions combines holes, asymptotes and intercepts into one sketch. Transforming the reciprocal graph handles any rule of the form \displaystyle y = \frac{a}{x - h} + k with a \ne 0, moving both asymptotes of \displaystyle y = \frac{1}{x}.
Applications. Rational functions in context models average cost, concentration and shared work, where one quantity is divided by another that changes.
A path through it
Asymptotes come before graphing a rational function, and the domain comes before working in context. Transforming the reciprocal graph can be taken at any point once you have the parent graph and graphing with transformations.
Next, polynomial and rational inequalities use the graph’s breaks and zeros as a sign chart, and calculus revisits the asymptotes through infinite limits and limits at infinity.