The fraction (ax + b)/(cx + d): domain and graph

What it shows

A rational expression such as (ax + b)/(cx + d) has a value only where its denominator is not zero, so its domain excludes x = −d/c. Graphing the expression turns this rule into a visible break in the curve. Near the excluded value the denominator is very small and the value grows without bound, giving a vertical asymptote; for large |x| the value settles towards a/c, the horizontal asymptote. When ad − bc = 0 the numerator is a multiple of the denominator, so the expression cancels to a constant and the graph becomes a horizontal line with one missing point.

How to use

Start with the default coefficients and drag point P along the graph, reading the value of the expression at several x-values, including some close to the break. Change Coefficient c and Coefficient d and ask students to predict where the break will move before they look, then turn on Show asymptotes to check. Next choose a, b, c, d with ad − bc = 0, for example 2, 4, 1, 2, and look for the open point.

Parameters you can change

• Coefficient a (numerator) -6–6
• Coefficient b (numerator) -6–6
• Coefficient c (denominator) -6–6
• Coefficient d (denominator) -6–6
• Show asymptotes
• Axis range (−R … R) 4–20

Questions to explore

1. Which value of x must be excluded from the domain of (x + 2)/(x − 1), and what does the graph do there?
2. Why can the graph never cross the vertical line x = −d/c?
3. What kind of expression, and what kind of graph, do you get when Coefficient c is set to 0?