Asymptotes and rational functions: vertical, horizontal and oblique asymptotes, holes, and how to read the whole graph from the algebra

A rational function is one polynomial over another, and its graph is decided by four things you read off the factored form: holes where a factor cancels, vertical asymptotes where the bottom is still zero after cancelling, a horizontal asymptote decided by comparing the degrees of top and bottom, and an oblique asymptote when the top's degree is exactly one more. Students lose the mark by calling every zero of the bottom a vertical asymptote. Here is one function read line by line, and a sibling to try.

Rational functions and asymptotes: the four-step reading

Every rational function is read in the same four steps, in this order.

1. Factor the top and the bottom completely.
2. Cancel any factor they share. Each cancelled factor marks a hole.
3. Read the vertical asymptotes from the zeros still left in the bottom.
4. Compare the degrees of top and bottom for the end behaviour: a horizontal asymptote, an oblique one, or neither.

The order matters. If you look for vertical asymptotes before cancelling, you will find one that is not there.

In MHF4U this is idea 7, "Rational functions, vertical and horizontal asymptotes", followed by idea 8, "Solving rational equations and inequalities". Before them comes idea 6, "Solving polynomial inequalities". After them the course moves to its exponential strand with idea 9, "Exponential functions and their graphs". The MHF4U course map has the full order, and the precalculus ideas page explains why calculus leans on this reading.

How to find vertical asymptotes, and a hole in a graph

Both come from the same place: an x that makes the denominator zero. What happens next decides which one you get.

• If the factor cancels with the top, the function is undefined at that x but stays finite near it. The graph has a single missing point: a hole. Its height is the simplified function's value at that x.
• If the factor does not cancel, the bottom goes to zero while the top does not. The function's size grows without bound. That is a vertical asymptote.

So a zero of the denominator is a question, not an answer. Cancelling tells you which it is.

Worked example: one rational function, line by line

Read f(x) = (2x² − 2)/(x² − x − 2) completely.

1. Factor. Top: 2x² − 2 = 2(x² − 1) = 2(x − 1)(x + 1). Bottom: x² − x − 2 = (x − 2)(x + 1).
2. Cancel the shared factor (x + 1). The simplified form is f(x) = 2(x − 1)/(x − 2), for x ≠ −1. There is a hole at x = −1.
3. Find the hole's height. Put x = −1 into the simplified form: 2(−1 − 1)/(−1 − 2) = (−4)/(−3) = 4/3. The hole is at (−1, 4/3).
4. Read the vertical asymptote. The bottom still has (x − 2), so there is a vertical asymptote at x = 2.
5. Compare degrees. Top and bottom are both degree 2, so the horizontal asymptote is the ratio of the leading coefficients: y = 2/1 = 2.
6. Find the intercepts. The x-intercept comes from the remaining top: x − 1 = 0, so x = 1. The y-intercept is f(0) = (−2)/(−2) = 1.
7. Test each side of the vertical asymptote. At x = 3: 2(2)/1 = 4, positive. At x = 1.5: 2(0.5)/(−0.5) = −2, negative. So the graph falls toward negative infinity just left of x = 2 and comes down from positive infinity just right of it.

That is the whole graph, read from the algebra before a single point is plotted.

The common wrong line, first version: "two vertical asymptotes, x = 2 and x = −1." At x = −1 the factor cancelled, and the function approaches 4/3. Nothing blows up. It is a hole.

The common wrong line, second version: "horizontal asymptote y = 0, because it is a fraction." y = 0 is the case where the bottom's degree is bigger. Here the degrees are equal, so the leading coefficients decide.

Booth and colleagues (2013) found that studying incorrect examples with the fault explained may be especially useful for understanding in algebra. That is why both wrong lines are written out in full.

Horizontal asymptote: the degree rule

How to find horizontal asymptotes comes down to comparing two numbers: the degree of the top and the degree of the bottom.

Degrees	End behaviour	Example
Top smaller than bottom	Horizontal asymptote y = 0	3x/(x² + 1)
Top equal to bottom	y = ratio of leading coefficients	(2x² − 2)/(x² − x − 2) has y = 2
Top one more than bottom	No horizontal; an oblique asymptote	(x² + 1)/(x − 1)
Top two or more higher	Neither	x³/(x − 1)

Why it works: for very large x, the leading terms swamp everything else. (2x² − 2)/(x² − x − 2) behaves like 2x²/x² = 2. This is a limit at infinity, which the limits page makes formal.

Oblique asymptote: divide

When the top's degree is exactly one more than the bottom's, the end behaviour is a slanted line. Ontario calls it an oblique asymptote; many US books call it a slant asymptote.

Take g(x) = (x² + 1)/(x − 1). Divide the top by the bottom. The quotient is x + 1 and the remainder is 2, so:

g(x) = x + 1 + 2/(x − 1).

Check the division: (x + 1)(x − 1) + 2 = x² − 1 + 2 = x² + 1.

For large x, the term 2/(x − 1) shrinks to zero, so the graph hugs the line y = x + 1. That line is the oblique asymptote. The bottom is zero at x = 1, with nothing to cancel, so there is also a vertical asymptote at x = 1. The division itself is worked step by step on the polynomial division page.

Sibling to try cold

Read k(x) = (x² − 9)/(x² − 4x + 3) completely: holes, vertical asymptotes, horizontal asymptote and intercepts. Cover the example above and write every line. A worked example followed by your own attempt is the order Sweller and Cooper (1985) found teaches beginners fastest.

Check

The factored form is (x − 3)(x + 3)/((x − 1)(x − 3)). Cancel (x − 3): a hole at x = 3, with height (3 + 3)/(3 − 1) = 3, so the hole is at (3, 3). The vertical asymptote is x = 1. The degrees are equal and both leading coefficients are 1, so the horizontal asymptote is y = 1. The x-intercept is x = −3, and the y-intercept is k(0) = (−9)/3 = −3.

If you wrote two vertical asymptotes, you looked before you cancelled.

From the reading to the sketch

There is no picture on this page, so here is the drawing order to follow on paper.

1. Draw every asymptote as a dashed line.
2. Mark each hole as an open circle at its height.
3. Mark the intercepts.
4. Pick one test point in each region between the vertical asymptotes and the x-intercepts, and mark whether the graph is above or below the axis there.
5. Draw the curve through the points, bending toward each asymptote without touching a vertical one.

Rational equations and inequalities, in one paragraph

To solve a rational equation, multiply through by the common denominator, solve the polynomial that is left, then reject any root that makes an original denominator zero. It is the same rejected-root habit as in logarithmic equations. For a rational inequality, mark every zero of the top and the bottom on a number line and test the sign in each interval. The zeros of the bottom are never included, because the function is undefined there.

Where it goes next

In MCV4U, idea 15, "Curve sketching with derivatives and asymptotes", adds turning points and concavity to this reading. The degree rule becomes a limit at infinity in calculus. Practise the four steps on paper with the page closed; testing yourself before you reread explains why the closed page matters, and worked examples, then practice problems sets out the order. When the hole-versus-asymptote mistake keeps returning, chapter 4 of the free playbook is about finding that line. The other MHF4U ideas are on the math help hub.

Asymptotes on one screen

• Four steps: factor, cancel, read the remaining zeros of the bottom, compare degrees.
• Cancel means hole, remain means asymptote. A hole's height is the simplified function at that x.
• Degree rule: top smaller gives y = 0; equal gives the leading coefficients' ratio; one more gives an oblique asymptote; two or more gives neither.
• Oblique asymptote: divide; the quotient is the line.
• Sketch: dashed asymptotes, open-circle holes, intercepts, one test point per region.
• Equations: multiply through, solve, reject any root that zeroes a denominator.