Rational functions are a powerful way to model relationships where one quantity depends on the reciprocal of another (for example, "as input gets close to a certain value, output grows without bound"). In many applications this kind of behaviour appears naturally, and graphs help you see it immediately through asymptotes.
Definition: Rational Function
A rational function is any function that can be written as a quotient $\frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials and $Q(x)\neq 0$.
The condition $Q(x)\neq 0$ is not a minor detail. It creates restrictions on the domain, and those restrictions often show up on the graph as vertical asymptotes (or sometimes holes, in more advanced cases).
A key "parent" example is the reciprocal function:
Definition: Reciprocal Function
The reciprocal function is $f(x)=\frac{1}{x}$ with $x\neq 0$.
Its domain excludes 0 because division by 0 is undefined.
In many textbooks, the word "reciprocal" refers to the multiplicative inverse of a number (for example, the reciprocal of 5 is $\frac{1}{5}$). The reciprocal function $y=\frac{1}{x}$ turns each input $x$ into its multiplicative inverse, but that is different from the idea of an inverse function (which reverses a whole input-output process).
The graph of $y=\frac{1}{x}$ has two separate curves (often called branches): one in Quadrant I and one in Quadrant III.
As $x$ gets closer to 0:
So the graph gets closer and closer to the vertical line $x=0$ but never touches it.
Definition: Vertical Asymptote
A vertical asymptote is a vertical line $x=c$ that a graph approaches as the input approaches $c$. For rational functions, it often occurs where the denominator approaches 0.
For $y=\frac{1}{x}$, the vertical asymptote is $x=0$.
As $x\to \pm\infty$, $\frac{1}{x}\to 0$. The graph approaches, but does not reach, the horizontal line $y=0$.
Definition: Horizontal Asymptote
A horizontal asymptote is a horizontal line $y=c$ that a graph approaches as $x\to +\infty$ and/or $x\to -\infty$.
For $y=\frac{1}{x}$, the horizontal asymptote is $y=0$.
For $y=\frac{1}{x}$:
For the basic reciprocal shape, "excluded $x$-value" corresponds to a vertical asymptote, and "excluded $y$-value" corresponds to a horizontal asymptote.
A very important rational family is
$$f(x)=\frac{a}{x-h}+k \quad (a\neq 0).$$
You can think of this as transforming the parent function $\frac{1}{x}$ using:
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