Analyze and graph rational functions whose numerator and denominator each have degree at most 2, including poles, removable holes and horizontal, slant or polynomial end behavior.
Each expression is resolved exactly before drawing. The key distinguishes remaining poles from fully canceled holes, excludes original denominator zeros from the intercepts and shows the correct horizontal, slant or polynomial end behavior.
The set uses useful classroom cases with nonzero numerators through degree 2 and nonconstant denominators through degree 2. It includes simple and repeated poles, two distinct poles, removable holes, no-real-pole cases and full cancellation to a polynomial.
Questions teachers ask
What the key records
The key lists original domain exclusions, the reduced expression, holes, intercepts and end behavior before showing the graph. Curves are split at poles, holes stay open and asymptotes are dashed.
Learning objective
What this activity builds
Use factoring, original domain exclusions, intercepts and end behavior to sketch a rational function accurately.
Worked example
What happens to f(x) = (x² - 1)/(x - 1) at x = 1?
The reduced rule is y = x + 1, with a hole at (1, 2).
The factor x - 1 cancels, but x = 1 was excluded by the original denominator. Plot the line and leave that point open.
Teaching notes
Make the practice count
List original denominator zeros before canceling factors.
Classify each excluded value as a hole or a pole after reduction.
Use polynomial division to identify horizontal, slant or polynomial end behavior.
Plot each interval between poles as its own branch.
Make it yours
Customize the practice
Choose your options, update the preview, then download the exact same pages.
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