Cogito
Integrated Math 3 · Chapter 2 · Lesson 1
Rational Functions and Asymptotes
Where a function is undefined.
12 problems · about 21 minutes · F-IF.C.7, A-APR.D.6
What this lesson teaches
The student identifies vertical and horizontal asymptotes and holes of a rational function.
- A rational function is undefined where its denominator is zero.
- A cancelling factor gives a hole; otherwise there is a vertical asymptote.
- Compare degrees for the horizontal asymptote.
Warm Up
Straightforward practice. Get the method working first.
5 problemsf(x) = (8x + 3) ÷ (4x − 1). What is the horizontal asymptote value?
Answer 2
Why 2.
What produces a hole rather than an asymptote?
Answer A factor that cancels from both top and bottom.
Why A shared, cancelling factor.
f(x) = 5 ÷ (x + 2). Vertical asymptote at which x?
Answer -2
Why x + 2 = 0.
f(x) = (4x) ÷ (x + 1). Horizontal asymptote value?
Answer 4
Why 4 ÷ 1.
f(x) = (x + 2) ÷ (x² + 1). Horizontal asymptote value?
Answer 0
Why The bottom degree is larger.
Build It Up
The same ideas with more to keep track of.
3 problemsf(x) = (x² − 4) ÷ (x − 2). Asymptote or hole at x = 2? 1 asymptote, 2 hole.
Answer 2
Why The factor cancels.
f(x) = (9x²) ÷ (3x² + 1). Horizontal asymptote value?
Answer 3
Why 9 ÷ 3.
f(x) = 1 ÷ x. Horizontal asymptote value?
Answer 0
Why The bottom grows without bound.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsSort each situation by what appears at that x value.
Answer Vertical asymptote: Denominator zero, numerator not, A factor only in the denominator · Hole: A factor cancelling from both, Both top and bottom zero from the same factor
Why Cancelling is what makes a hole.
f(x) = 3 ÷ (x − 8). Vertical asymptote at which x?
Answer 8
Why x − 8 = 0.
The Break: f(x) = 1 ÷ (x − 7). At which x is the vertical asymptote?
Answer 7
Why x = 7.
The Far End: f(x) = (6x + 1) ÷ (2x − 5). What is the horizontal asymptote value?
Answer 3
Why 3.