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.
Understanding Math: Asymptotes of Rational Functions
Near a vertical asymptote, the important idea is not just that the function is undefined at one input. The outputs can grow without bound as the input gets close to that value. A graph may rise toward positive infinity on one side and fall toward negative infinity on the other.
In some cases, both sides rise or both sides fall. You can predict this by checking the signs of the factors just to the left and right of the restricted input. For example, a denominator factor that occurs once often changes sign across its zero.
A factor that occurs twice usually keeps the same sign. This sign check prevents sketches that put both branches on the wrong side of the graph.
A hole behaves very differently from a vertical asymptote. After a common factor is removed, the simplified rule tells where the nearby points lie, but the original function still has one missing point. Find its height by using the simplified rule at the excluded input.
On a graph, mark that location with an open circle. This matters because two expressions can produce the same output for nearly every input while still define different functions.
In algebra, cancellation is valid only when the canceled factor is not zero. Keeping track of that restriction is a useful habit in later work with fractions, equations, and limits.
End behavior comes from comparing which polynomial grows faster when inputs become very large in either direction. Lower degree terms matter less and less far from the origin. That is why a horizontal asymptote can often be found from only the leading terms.
A horizontal asymptote is not a wall. The graph may cross it once, many times, or not at all. A slant asymptote gives a tilted line instead.
Polynomial division separates the function into a quotient and a leftover fraction. Far from the origin, that leftover fraction becomes close to zero, so the quotient describes the graph's direction.
The graph can cross a slant asymptote too. If the numerator degree is more than one greater, division can produce a curved polynomial asymptote rather than a line.
When sketching, start with the domain restrictions before making a table of values. Mark holes and vertical asymptotes, then draw the horizontal or slant asymptote as a guide. Choose test inputs in every interval created by the vertical asymptotes.
Include inputs close to each side, since ordinary table values can hide rapid changes. Check intercepts only when they are allowed by the domain. A numerator zero gives an input intercept unless that zero was canceled.
Finally, compare the signs and the end behavior with your sketch. Rational graphs often look complicated, but each branch follows a small set of rules. Careful factoring and a few well chosen test values reveal most of the shape.
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.