Rational Functions and Asymptotes

Where a function is undefined.

12 problems · about 21 minutes · F-IF.C.7, A-APR.D.6

NameDate

Figure — use these to answer the problems

Follow the curve toward zero and it runs away. Follow it outward and it flattens toward the axis.

y = 2/x + 0

Point on the curve3

Figure 1

Straightforward practice. Get the method working first.

5 problems

f(x) = (8x + 3) ÷ (4x − 1). What is the horizontal asymptote value?
Answer
2
What produces a hole rather than an asymptote?
- A factor that cancels from both top and bottom.
- A small denominator.
f(x) = 5 ÷ (x + 2). Vertical asymptote at which x?
Answer
f(x) = (4x) ÷ (x + 1). Horizontal asymptote value?
Answer
f(x) = (x + 2) ÷ (x² + 1). Horizontal asymptote value?
Answer

The same ideas with more to keep track of.

3 problems

f(x) = (x² − 4) ÷ (x − 2). Asymptote or hole at x = 2? 1 asymptote, 2 hole.
Answer
f(x) = (9x²) ÷ (3x² + 1). Horizontal asymptote value?
Answer
f(x) = 1 ÷ x. Horizontal asymptote value?
Answer

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

4 problems

Sort each situation by what appears at that x value.
Write each item under the heading it belongs to: Denominator zero, numerator not · A factor only in the denominator · A factor cancelling from both · Both top and bottom zero from the same factor
Vertical asymptote
Hole
f(x) = 3 ÷ (x − 8). Vertical asymptote at which x?
Answer
The Break
f(x) = 1 ÷ (x − 7). At which x is the vertical asymptote?
Answer
The Far End
f(x) = (6x + 1) ÷ (2x − 5). What is the horizontal asymptote value?
Answer