Learning goal and prerequisites
Classify different failures and long-run features of a rational rule. Before starting, review rational functions.
Method and conditions
Record original denominator restrictions first. Cancel common factors only on the allowed domain. A surviving denominator zero can create a vertical asymptote; a canceled simple factor creates a hole when the reduced expression is finite there. Use division for a slant asymptote.
Worked example: show the decisions
R(x)=(x−2)(x+3)/[(x−2)(x−5)] equals (x+3)/(x−5) except at 2 and 5. At x=2 the reduced value is −5/3, locating the hole. At x=5 the numerator remains 8, so the vertical asymptote is x=5.
Contrast: why a plausible shortcut fails
A zero denominator alone does not distinguish a hole from an asymptote. For (x²+3x+1)/(x+1), division gives x+2−1/(x+1), so the slant asymptote is y=x+2, not y=1.
Transfer: try before revealing the answer
Analyze (x−1)(x+2)/[(x−1)(x−4)] and state both excluded inputs with their different graph meanings.
Transfer solution and check
x=1 is a hole at y=−1; x=4 is a vertical asymptote. The equal leading degrees give horizontal asymptote y=1.
R(x)=(x−3)(x+1)/[(x−3)(x−5)]. What is the x-coordinate of its vertical asymptote?
- Cancel the common factor while retaining its hole; the surviving denominator zero gives the asymptote.
- Using these quantities: 5 = 5.
- Check the original domain, requested quantity and units; do not substitute a related parameter.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
AP Precalculus Course at a Glance