Functions - Rational Functions

Definition and Domain: A rational function is defined as the quotient of two polynomial functions, expressed as $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$. The domain consists of all real numbers except those that make the denominator $Q(x) = 0$. Visually, these points of exclusion appear as breaks or gaps in the graph, often represented by vertical dashed lines.

Vertical Asymptotes (VA): These occur at the $x$-values where the denominator is zero (and the numerator is non-zero). As the input $x$ approaches this value, the output $f(x)$ shoots toward positive or negative infinity. On a coordinate plane, the graph will curve sharply to follow a vertical dashed line $x = c$ without ever crossing it.

Horizontal Asymptotes (HA): These lines $y = L$ indicate the value the function approaches as $x$ moves toward positive or negative infinity. For a linear fractional function $f(x) = \frac{ax+b}{cx+d}$, the HA is the line $y = \frac{a}{c}$. Visually, this is a horizontal boundary that the 'tails' of the graph level off toward at the far left and right edges.

Intercepts: The $y$-intercept is the point where the graph crosses the vertical axis, found by calculating $f(0)$. The $x$-intercepts (roots) are the points where the graph crosses the horizontal axis, found by setting the numerator $P(x) = 0$. These points provide specific anchors for sketching the curve.

The Reciprocal Parent Function: The most basic rational function is $f(x) = \frac{1}{x}$. Its graph is a hyperbola with two symmetrical branches located in the first and third quadrants. It is an odd function (rotational symmetry about the origin) with a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$.

Transformations of $\frac{1}{x}$: Using the form $f(x) = \frac{a}{x-h} + k$, we can describe any transformation. The parameter $h$ shifts the graph (and the VA) horizontally, $k$ shifts the graph (and the HA) vertically, and $a$ determines the vertical stretch. If $a$ is negative, the graph is reflected over the $x$-axis, moving the branches into the second and fourth quadrants.

Slant (Oblique) Asymptotes: These occur when the degree of the numerator is exactly one higher than the degree of the denominator. Instead of leveling off horizontally, the graph approaches a diagonal line $y = mx + c$ as $x \to \pm\infty$. This is found using polynomial long division.