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Precalculus Unit 3: Rational functions

Rational functions are ratios of polynomials. Their graphs are shaped by asymptotes and holes.

Covers: Vertical/horizontal/slant asymptotes, holes, rational inequalities

Study guide

A rational function is a ratio of polynomials, r(x) = p(x)/q(x). Its behavior is controlled by the zeros of the numerator and denominator. Start by factoring both and canceling any common factors.

A factor that cancels creates a hole: a single missing point at that x-value. A factor left in the denominator after canceling creates a vertical asymptote, where the function grows without bound near that x. Zeros of the reduced numerator are the x-intercepts.

End behavior depends on degrees. If the numerator's degree is less than the denominator's, the horizontal asymptote is y = 0. If the degrees are equal, it's y = (ratio of leading coefficients). If the numerator's degree is exactly one more, there's a slant (oblique) asymptote: divide the polynomials, and the quotient without its remainder is the asymptote's equation. If the numerator's degree is two or more higher, the end behavior follows the quotient polynomial.

Rational inequalities work like polynomial inequalities, except the sign can change at both zeros and vertical asymptotes. Move everything to one side as a single fraction, mark the zeros of the numerator and denominator on a number line, test each interval, and never include x-values that make the denominator zero.

Near a vertical asymptote x = a, describe the behavior with one-sided limits, for example lim(x→a⁺) r(x) = −∞, using the sign of r just to the right of a. A horizontal asymptote y = L means lim(x→±∞) r(x) = L. To locate a hole, cancel the common factor, then substitute its x-value into the simplified function. The result is the y-coordinate of the missing point.

A rational expression with a factored denominator can be split into simpler fractions, such as (5x − 1)/((x − 1)(x + 1)) = A/(x − 1) + B/(x + 1). Multiply through by the denominator, then substitute convenient x-values (or match coefficients) to find A and B.

Key ideas

• Factor and reduce first. Canceled factors create holes.
• Vertical asymptotes come from zeros of the reduced denominator.
• Horizontal asymptote: degree bottom > top → y = 0; equal → ratio of leading coefficients.
• If the top's degree is exactly one more, there's a slant asymptote found by division.
• Near a vertical asymptote, the function heads to +∞ or −∞.

Formulas and rules

Common mistakes

• Calling a hole a vertical asymptote.
• Finding a horizontal asymptote when the top's degree is bigger.

Words to know

Hole (rational function)

A removable point at a zero of a factor that cancels.

Horizontal asymptote (equal degrees)

y = ratio of the leading coefficients.

Slant asymptote

Occurs when the top's degree is exactly one more than the bottom's.

Finding a slant asymptote

Divide the polynomials; the quotient (no remainder) is the line.

Vertical asymptote

A line x = a where the reduced denominator is 0.

Rational function

A ratio of polynomials p(x)/q(x).

Horizontal asymptote (bottom degree bigger)

y = 0

Oblique asymptote

Another name for a slant asymptote.

X-intercepts of a rational function

Zeros of the reduced numerator.

Rational inequality

Sign changes can happen at numerator zeros and denominator zeros.

Sample practice questions

21 more questions like these, from level 1 to 5, are in the practice set.