Rational Functions

Asymptotes, holes, and the forbidden denominators.

Interactive graph

Interactive Lab

Drag corner to resize ↘

Drag the sliders in the Desmos graph. Watch how each parameter reshapes the curve in real time.

VisualThe reciprocal function — a hyperbola hugging both axes, never touching them.

Algebra $y = \tfrac{1}{x}$ . Vertical asymptote: $x=0$ . Horizontal asymptote: $y=0$ .

y = \frac{1}{x}

VisualThe graph sprints toward infinity at the forbidden x-value. Find where the denominator is zero — but the numerator isn't.

AlgebraSet denominator $=0$ (and verify numerator $\neq 0$ ). Drag h to slide the red vertical line.

x = h

VisualThe resting height the curve approaches as x runs to ±∞. Drag k to lift it.

AlgebraCompare degrees of numerator and denominator. For $\tfrac{a}{x-h}+k$ , it's $y=k$ .

y = k

VisualA pinpoint gap — the curve approaches a real point but never reaches it. Happens when a factor cancels.

AlgebraIf $(x-r)$ divides both numerator and denominator, there's a hole at $x=r$ .

Reciprocal parent

y = \frac{1}{x}

Hyperbola, asymptotes on axes

Shifted rational

y = \frac{a}{x-h} + k

VA: x=h, HA: y=k

Hole location

x = r \text{ where factor cancels}

Compute y by limit

Mark this lesson complete to update your mastery.

easyrationals

Find the vertical asymptote of

y=\dfrac{1}{x-5}

easyrationals

Find the horizontal asymptote of

y=\dfrac{3}{x}+2

mediumrationals

Find all asymptotes of

y=\dfrac{2x+1}{x-3}

mediumrationals

Does

y=\dfrac{x^2-4}{x-2}

have a hole or a vertical asymptote at

x=2

hardrationals

Find the hole(s) of

y=\dfrac{x^2-5x+6}{x^2-4x+3}