Rational functions and asymptotes

Learning intention

Analyze rational functions by connecting algebraic form, asymptotes, intercepts and graphical behavior.

The core idea

A rational function is a quotient of polynomials. Values that make the denominator zero are excluded from the domain, but an excluded value may create either a vertical asymptote or a removable hole.

End behavior comes from comparing polynomial degrees or performing division. The graph should agree with the algebra: intercepts, signs and asymptotes together determine its main branches.

Key definitions and formulas

Vertical asymptote: x = a when an uncancelled denominator factor is zero.
Horizontal asymptote: compare leading terms when numerator degree is no greater than denominator degree.
Oblique asymptote: the quotient from polynomial division when numerator degree is one greater.
Hole: an excluded value from a factor that cancels.

Worked example 1 — foundation

Analyze f(x) = 1/(x − 2) + 3.

The denominator is zero at x = 2, giving vertical asymptote x = 2.
As |x| grows, 1/(x − 2) approaches 0, giving horizontal asymptote y = 3.
The x-intercept solves 1/(x − 2) = −3, so x = 5/3.

The graph is a translated reciprocal curve centered around (2, 3).

Worked example 2 — exam-style

Analyze g(x) = (2x² + x − 3)/(x − 1).

Factor the numerator: (2x + 3)(x − 1).
For x ≠ 1, g(x) = 2x + 3.
Because the factor cancels, x = 1 is a hole, not a vertical asymptote.
The missing point would be (1, 5).

The graph is the line y = 2x + 3 with an open point at (1, 5).

Worked example 3 — HL reasoning

Find the oblique asymptote of h(x) = (x² + 3x + 5)/(x + 1).

Divide x² + 3x + 5 by x + 1.
The quotient is x + 2 and the remainder is 3.
Write h(x) = x + 2 + 3/(x + 1).
The remainder term approaches 0 as |x| grows.

The oblique asymptote is y = x + 2.

GDC checkpoint

Common IB mistake

A denominator zero does not automatically create a vertical asymptote. Factor first: a common factor may cancel and leave a removable discontinuity.

Practice checks

Use the feedback to refine your method, not only your final answer.

{"questions":[{"content":"What is the vertical asymptote of f(x) = 4/(x + 3)?<br>[[choice-rat-l1]]","widgets":{"choice-rat-l1":{"type":"choice","options":["x = −3","y = −3","x = 4","y = 4"],"answer":[0],"explanations":["x + 3 = 0 gives x = −3.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Set the denominator equal to zero."],"explanation":"x + 3 = 0 gives x = −3."},{"content":"What is the horizontal asymptote of (3x + 1)/(x − 2)?<br>[[choice-rat-l2]]","widgets":{"choice-rat-l2":{"type":"choice","options":["y = 3","x = 3","y = 1","There is none"],"answer":[0],"explanations":["Equal degrees give y = 3/1 = 3.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Compare leading coefficients."],"explanation":"Equal degrees give y = 3/1 = 3."},{"content":"The factor x − 5 cancels from numerator and denominator. What usually occurs at x = 5?<br>[[choice-rat-l3]]","widgets":{"choice-rat-l3":{"type":"choice","options":["A removable hole","A horizontal asymptote","A turning point","No domain restriction"],"answer":[0],"explanations":["Cancellation leaves a hole while x = 5 remains excluded.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["The original denominator is still zero there."],"explanation":"Cancellation leaves a hole while x = 5 remains excluded."},{"content":"For f(x) = 2 + 5/(x − 1), the horizontal asymptote is y = [[input-rat-l4]]","widgets":{"input-rat-l4":{"type":"input","answer":"2"}},"hints":["The fraction approaches zero."],"explanation":"The translated reciprocal approaches y = 2."},{"content":"Which method reveals an oblique asymptote?<br>[[choice-rat-l5]]","widgets":{"choice-rat-l5":{"type":"choice","options":["Polynomial division","Completing the square only","Differentiating the denominator only","A frequency table"],"answer":[0],"explanations":["Polynomial division separates the linear quotient from a vanishing remainder.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again.","Check the mathematical conditions and try again."]}},"hints":["Rewrite the rational expression as quotient plus remainder."],"explanation":"Polynomial division separates the linear quotient from a vanishing remainder."}],"mix":1}

Summary

Factor before classifying excluded values.
Use degree comparison or division for end behavior.
Combine algebraic features with a carefully chosen graphing window.
State asymptotes as equations.

Learning intention

The core idea

Key definitions and formulas

Worked example 1 — foundation

Worked example 2 — exam-style

Worked example 3 — HL reasoning

GDC checkpoint

Common IB mistake

Practice checks

Summary

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Rational functions and asymptotes

Learning intention

The core idea

Key definitions and formulas

Worked example 1 — foundation

Worked example 2 — exam-style

Worked example 3 — HL reasoning

GDC checkpoint

Common IB mistake

Practice checks

Summary

Rate this lesson

What can be improved?

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What needs to be fixed?

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