Asymptotes Practice

Asymptotesare lines that the graph of a rational function may approach.

  • Vertical asymptotes occur at values for x at which the domain is undefined.

Look for division by zero!

  • Horizontal asymptotes describe the end behavior of some rational functions.
  • Oblique (slant) asymptotes describe the end behavior of some rational functions.

If the expression that a graph approaches as x ±∞ is not a line, the rational function doesn’t have any horizontal or oblique asymptotes.

Use the graph to find the following.

Example A) / a) Domain: x ≠ -1, x ≠ 3
Range: (-∞, ∞)
b) Intercepts, if any:
x-int:x = -3, 1, 4 y-int:y = -5
c) Horizontal Asymptotes, if any:
y = 2.5
d) Vertical Asymptotes, if any:
x = -1 and x = 3
e) Oblique Asymptotes, if any:
none
Try #1 / 1a) Domain: ______
Range:______
1b) Intercepts, if any:
x-int:______y-int:______
1c) Horizontal Asymptotes, if any:
1d) Vertical Asymptotes, if any:
1e) Oblique Asymptotes, if any:

Horizontal vs ObliqueAsymptotes: Look at the degree of the numerator and denominator!

DegreeN < DegreeD
HA at y = 0 / DegreeN = DegreeD
HA at y = ratio of leading coefficients / DegreeN = DegreeD + 1
OA at y = quotient of the numerator and denominator / DegreeN > DegreeD + 1
No HA or OA

Find the equation of the oblique/slant asymptote.

Example B) / 2.
3. / 4.

Which of the following is NOT an asymptote of the function’s graph?

Ex C)
A y = 0
B x = 2
C x = 3
D y = 3 / 5.
A x = 5
B x = 1
C y = 0
D y = x – 2 / 6.
A y = 0
B x = 3
C x = 4
D y = 3x + 5

Find the vertical, horizontal and oblique asymptotes, if any, for each function.

7. / 8.