Rational functions – asymptotes of the graph

What it shows

An asymptote is a line that a graph approaches ever more closely. For y = (ax + b)/(cx + d), a root of the denominator gives a vertical asymptote x = −d/c unless the numerator is also zero there; as x → ±∞ the ratio tends to a/c. For y = (ax² + bx + c)/(x + d), polynomial division gives an oblique asymptote. A common factor (x − k) in the numerator and denominator cancels, yet x = k stays excluded from the domain, so the graph has a hole there instead of an asymptote. The workbench factors, cancels, adds, subtracts, multiplies and divides algebraic fractions over their common domain.

How to use

On the Graph & asymptotes tab, change the coefficients and predict the asymptotes before reading the panel. Tick Add common factor (x − k) and move k: the table shows y approaching the same value from both sides of the hole. Then try the task. On the Fraction workbench tab, choose the numerators, denominators and operation, and reveal the working with Next step.

Parameters you can change

• Function type (ax + b)/(cx + d), (ax² + bx + c)/(x + d)
• Coefficient a -5–5
• Coefficient b -8–8
• Coefficient c (linear denominator) / constant term c -5–5
• Coefficient d -6–6
• Screen Graph & asymptotes, Fraction workbench
• Common factor (x − k) in numerator and denominator
• Root k of the common factor -6–6
• First numerator 1, x, x − 1, x + 1, x − 2, x + 2, x − 3, x + 3, 2x + 4, 3x − 6, x², x² − x, x² + 2x, x² − 1, x² − 4, x² − 9, x² + x − 2, x² − x − 2, x² − 3x + 2, x² − 5x + 6, x² + 4x + 4, x² − 2x + 1, x² + 1, 2x² − 8
• First denominator 1, x, x − 1, x + 1, x − 2, x + 2, x − 3, x + 3, 2x + 4, 3x − 6, x², x² − x, x² + 2x, x² − 1, x² − 4, x² − 9, x² + x − 2, x² − x − 2, x² − 3x + 2, x² − 5x + 6, x² + 4x + 4, x² − 2x + 1, x² + 1, 2x² − 8
• Operation Simplify one fraction, Add, Subtract, Multiply, Divide
• Second numerator 1, x, x − 1, x + 1, x − 2, x + 2, x − 3, x + 3, 2x + 4, 3x − 6, x², x² − x, x² + 2x, x² − 1, x² − 4, x² − 9, x² + x − 2, x² − x − 2, x² − 3x + 2, x² − 5x + 6, x² + 4x + 4, x² − 2x + 1, x² + 1, 2x² − 8
• Second denominator 1, x, x − 1, x + 1, x − 2, x + 2, x − 3, x + 3, 2x + 4, 3x − 6, x², x² − x, x² + 2x, x² − 1, x² − 4, x² − 9, x² + x − 2, x² − x − 2, x² − 3x + 2, x² − 5x + 6, x² + 4x + 4, x² − 2x + 1, x² + 1, 2x² − 8
• Hide the steps so students work first

Questions to explore

1. Does a root of the denominator always give a vertical asymptote, or can it give a hole instead?
2. What is the oblique asymptote of y = (x² + 1)/(x − 2)?
3. Can the graph of y = (ax + b)/(cx + d) ever cross its horizontal asymptote y = a/c? Why?