Rational Functions & Asymptotes
A rational function is a ratio of two polynomials — a fraction where the numerator and denominator are both polynomial expressions. These functions produce some of the most dramatic graph behavior in precalculus: curves that shoot off to infinity, invisible barriers the function can never cross, and holes that are there but not there.
The Parent Function: y = 1/x
The simplest rational function is y = 1/x. It defines the shape that all other rational functions are built from.
Two key features:
- Vertical asymptote at x = 0: As x approaches 0 from the right, y shoots to positive infinity. From the left, y plunges to negative infinity. The function is undefined at x = 0 (you cannot divide by zero).
- Horizontal asymptote at y = 0: As x gets very large (positive or negative), 1/x gets closer and closer to zero but never reaches it.
Inverse relationships in physics follow this shape. The intensity of light decreases as 1/r^2 (distance squared). Gravitational force is proportional to 1/r^2. The curve y = 1/x is the template for “the farther away, the weaker.”
Shifting the Asymptotes: y = 1/(x - h) + k
By replacing x with (x - h) and adding k, we can move both asymptotes anywhere we want:
- h shifts the vertical asymptote to x = h.
- k shifts the horizontal asymptote to y = k.
The red horizontal line is the horizontal asymptote at y = k. The vertical asymptote is at x = h (you can see where the curve breaks). Drag the sliders and watch the entire curve slide around the plane.
Set h = 2 and k = 3. The curve now has its “center” at the point (2, 3) instead of the origin. The function approaches y = 3 as x goes to infinity, and it blows up near x = 2. Every rational function transformation follows this pattern.
Scaling and Reflection
Adding a coefficient a stretches and can flip the curve:
y = a / (x - h) + k
- |a| > 1: The curve stretches away from the asymptotes — the branches are “pushed out.”
- |a| < 1: The curve compresses toward the asymptotes.
- a < 0: The curve flips — branches that went up now go down, and vice versa.
Rational Functions with Polynomial Numerators
More interesting rational functions have polynomials of degree 1 or higher in both numerator and denominator. Let’s explore:
y = (x + n) / (x + d)
The horizontal asymptote is at y = 1 (the ratio of the leading coefficients, which are both 1). The vertical asymptote is at x = -d. The x-intercept is at x = -n.
Finding asymptotes from the equation:
- Vertical asymptote: Set the denominator equal to zero and solve.
- Horizontal asymptote: Compare the degrees of numerator and denominator. Same degree means HA = ratio of leading coefficients. Numerator degree less means HA = 0. Numerator degree greater means no horizontal asymptote (but possibly a slant asymptote).
End Behavior and Slant Asymptotes
When the numerator has degree exactly one more than the denominator, the function has a slant (oblique) asymptote instead of a horizontal one.
Consider y = (x^2 - 1) / x = x - 1/x. As x gets large, the -1/x part vanishes, so the function approaches the line y = x.
The curve hugs the yellow line y = x at the extremes but breaks away near the vertical asymptote at x = 0.
Holes in Rational Functions
A hole occurs when a factor cancels from both numerator and denominator. Consider y = (x^2 - 4) / (x - 2) = (x+2)(x-2) / (x-2). The (x-2) factors cancel, leaving y = x + 2, but with a hole at x = 2.
The two graphs look identical — and they are identical except at x = 2, where the original function is undefined. The hole is at the point (2, 4). It is invisible on the graph but algebraically present.
Challenge: Consider f(x) = (2x^2 + x - 6) / (x^2 - 4). Factor both numerator and denominator. Identify any holes and any vertical asymptotes. What is the horizontal asymptote? Sketch the graph mentally before checking with a graphing tool.