Slant Asymptotes and Rational Graphs

When the end behaviour is a line.

12 problems · about 21 minutes · AP Precalculus 1.12

NameDate

Figure — use these to answer the problems

The fractional part of a divided rational function behaves like this: it fades toward zero far from the origin.

y = 1/x + 0

Point on the curve3

Figure 1

Straightforward practice. Get the method working first.

5 problems

Numerator degree 3, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer
2
How do you find a slant asymptote?
- Divide, and take the polynomial part of the quotient.
- Take the ratio of the leading coefficients.
Numerator degree 2, denominator degree 1. Slant asymptote? 1 yes, 0 no.
Answer
Numerator degree 1, denominator degree 1. Slant asymptote? 1 yes, 0 no.
Answer
Numerator degree 4, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

Can a rational function have both a horizontal and a slant asymptote? 1 yes, 0 no.
Answer
f(x) = (6x + 1) ÷ (2x − 5). Horizontal asymptote value?
Answer
f(x) = 5 ÷ (x + 3). Vertical asymptote at which x?
Answer

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each degree comparison by the end behaviour it gives.
Write each item under the heading it belongs to: Equal degrees · Numerator degree smaller · Numerator exactly one higher · Degree 3 over degree 2
Horizontal asymptote
Slant asymptote
f(x) = (x + 2) ÷ (x² + 1). Horizontal asymptote value?
Answer
The Degrees
Numerator degree 3, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer
The Vertical
f(x) = (x² + 1) ÷ (x − 1). At which x is the vertical asymptote?
Answer