Cogito
AP Precalculus · Chapter 3 · Lesson 3
Slant Asymptotes and Rational Graphs
When the end behaviour is a line.
12 problems · about 21 minutes · AP Precalculus 1.12
What this lesson teaches
The student identifies slant asymptotes and sketches complete rational function graphs.
- A numerator degree exactly one higher gives a slant asymptote.
- Divide, and the polynomial part is the asymptote.
- A function has a horizontal or a slant asymptote, never both.
Warm Up
Straightforward practice. Get the method working first.
5 problemsNumerator degree 3, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer 1
Why Yes.
How do you find a slant asymptote?
Answer Divide, and take the polynomial part of the quotient.
Why Polynomial division.
Numerator degree 2, denominator degree 1. Slant asymptote? 1 yes, 0 no.
Answer 1
Why Exactly one higher.
Numerator degree 1, denominator degree 1. Slant asymptote? 1 yes, 0 no.
Answer 0
Why Equal degrees give a horizontal one.
Numerator degree 4, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer 0
Why A difference of two, not one.
Build It Up
The same ideas with more to keep track of.
3 problemsCan a rational function have both a horizontal and a slant asymptote? 1 yes, 0 no.
Answer 0
Why The degrees decide one or the other.
f(x) = (6x + 1) ÷ (2x − 5). Horizontal asymptote value?
Answer 3
Why 6 ÷ 2.
f(x) = 5 ÷ (x + 3). Vertical asymptote at which x?
Answer -3
Why x + 3 = 0.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each degree comparison by the end behaviour it gives.
Answer Horizontal asymptote: Equal degrees, Numerator degree smaller · Slant asymptote: Numerator exactly one higher, Degree 3 over degree 2
Why Only a difference of exactly one gives a slant.
f(x) = (x + 2) ÷ (x² + 1). Horizontal asymptote value?
Answer 0
Why The bottom degree is larger.
The Degrees: Numerator degree 3, denominator degree 2. Slant asymptote? 1 yes, 0 no.
Answer 1
Why Yes.
The Vertical: f(x) = (x² + 1) ÷ (x − 1). At which x is the vertical asymptote?
Answer 1
Why x = 1.