Rational functions are ratios of polynomials, and their graphs often have features that ordinary polynomial graphs do not. Asymptotes show where a graph approaches a line without necessarily touching it, which helps describe behavior near restricted x-values and far from the origin. Holes show removable breaks in the graph caused by factors that cancel.
Learning to identify these features makes rational function graphs much easier to sketch and interpret.
To analyze a rational function, first factor the numerator and denominator. Uncanceled denominator factors usually create vertical asymptotes, while canceled factors create holes. Horizontal and slant asymptotes describe end behavior and depend on the degrees of the numerator and denominator.
These tools let you predict the graph before plotting many points.
Key Facts
- A rational function has the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) is not zero.
- Vertical asymptotes occur at x-values that make the simplified denominator equal 0.
- A removable hole occurs when the same factor cancels from the numerator and denominator.
- If degree(P) < degree(Q), then the horizontal asymptote is y = 0.
- If degree(P) = degree(Q), then the horizontal asymptote is y = leading coefficient of P / leading coefficient of Q.
- If degree(P) = degree(Q) + 1, the slant asymptote is found by polynomial division: P(x)/Q(x) = quotient + remainder/Q(x).
Vocabulary
- Rational function
- A function that can be written as the quotient of two polynomials.
- Vertical asymptote
- A vertical line x = a that the graph approaches as x gets close to a from one or both sides.
- Horizontal asymptote
- A horizontal line y = b that describes the end behavior of a function as x approaches positive or negative infinity.
- Slant asymptote
- A nonhorizontal line that a rational function approaches when the numerator degree is exactly one more than the denominator degree.
- Removable hole
- A missing point in the graph caused by a factor that cancels from both the numerator and denominator.
Common Mistakes to Avoid
- Setting the original denominator equal to zero without simplifying first is wrong because canceled factors create holes, not vertical asymptotes.
- Forgetting to factor completely is wrong because hidden common factors can change the graph's holes and asymptotes.
- Using horizontal asymptote rules when the numerator degree is one more than the denominator degree is wrong because that case has a slant asymptote instead.
- Assuming the graph can never cross an asymptote is wrong because rational functions may cross horizontal or slant asymptotes, although they cannot pass through a vertical asymptote.
Practice Questions
- 1 Find the vertical asymptote and any removable hole for f(x) = (x^2 - 1)/(x^2 - 3x + 2).
- 2 Find the horizontal or slant asymptote of f(x) = (2x^2 + 5x - 1)/(x - 3).
- 3 Explain why f(x) = (x - 4)/(x^2 - 16) has a hole instead of a vertical asymptote at x = 4.