ASYMPTOTES OF RATIONAL FUNCTIONS WORKSHEET

Find the horizontal asymptote of the graph of each rational function.

Problem 1 :

y = 2/(x – 6)

Solution

Problem 2 :

y = (x + 2)/(x – 4)

Solution

Problem 3 :

y = (x + 3)/2(x + 4)

Solution

Problem 4 :

y = (2x² + 3)/(x² – 6)

Solution

Problem 5 :

y = (3x - 12)/(x² – 2)

Solution

Problem 6 :

y = (3x³ – 4x + 2)/(2x³ + 3)

Solution

For each function, determine the equations of any vertical asymptotes, the locations of any holes, and the existence of any horizontal or oblique asymptotes.

Problem 7 :

y = x/(x + 4)

Problem 8 :

y = 1/(x - 5) (x + 3)

Problem 9 :

y = (x + 4) / (x² - 16)

Problem 10 :

Consider the function

f(x) = 3/(x - 2)

a) State the equation of the vertical asymptote.

b) Use a table of values to determine the behaviour(s) of the function near its vertical asymptote.

c) State the equation of the horizontal asymptote. 

d) Use a table of values to determine the end behaviours of the function near its horizontal asymptote.

e) Determine the domain and range.

f ) Determine the positive and negative intervals.

g) Sketch the graph.

Answer Key

1)  y = 0

2)  y = 1

3)  y = 1/2

4)  y = 2

5)  y = 0 which is the x – axis.

6)  y = 1.5.

7) 

a) The vertical asymptote is at x = 2

b) The intervals are (-∞, 2) and (2, ∞)

• When x ∈ (-∞, 2), f(x) will be negative. That is, when x < 2, f(x) is negative.
• When x ∈ (2, ∞), f(x) will be positive. That is, when x > 2, f(x) is positive.

y-intercept is -3/2.

c) Highest exponent of the numerator = 0, highest exponent of the denominator = 1

Equation of horizontal asymptote is x-axis or y = 0.

d) End behavior :

• x --> -∞ then f(x) --> 0
• x --> ∞ then f(x) --> 0

e) Domain is all real numbers except x = 2

Range is all real values except y = 0

f) 

• x ∈ (-∞, 2), f(x) will be negative
• x ∈ (2, ∞), f(x) will be positive

Describe the vertical asymptotes and holes for the graph of each rational function.

Problem 1 :

y = (x - 2)/(x + 2) (x - 2)

Solution

Problem 2 :

y = x/x(x - 1)

Solution

Problem 3 :

y = (5 - x)/(x² - 1)

Solution

Problem 4 :

y = (x² - 2)/(x + 2)

Solution

Problem 5 :

y = (x² - 4)/(x² + 4)

Solution

Problem 6 :

y = (x + 3)/(x² - 9)

Solution

Problem 7 :

y = (x² - 25)/(x – 4)

Solution

Problem 8 :

y = (x - 2) (2x + 3)/(5x + 4) (x – 3)

Solution

Problem 9 :

y = (15x² - 7x - 2)/(x² - 4)

Solution

State the equation of the vertical asymptote of each function. Then choose a  strategy  to determine how the graph of the function approaches its vertical  asymptote

Problem 10 :

y = 2/(x + 3)

Problem 11 :

y = (x - 1)/(x - 5)

Find the oblique asymptote of the rational functions.

Problem 1 :

f(x) = (x² + 8x – 20)/(x – 1)

Solution

Problem 2 :

f(x) = (6x³ – 1)/(-2x² + 18)

Solution

Problem 3 :

f(x) = (2x² + x – 5)/(x + 1)

Solution

Problem 4 :

f(x) = (2x² - 5x + 3)/(x – 1)

Solution

Problem 5 :

f(x) = (2x² - 5x + 5)/(x – 2)

Solution

Problem 6 :

f(x) = (x³ - 2x² + 5)/x²

Solution

Problem 7 :

f(x) = (x³ - x² - x - 1)/(x – 3) (x + 4)

Solution

Problem 8 :

f(x) = x³/(x² – 4)

Solution

Problem 9 :

Without using graphing technology, match each equation with its corresponding graph. Explain your reasoning.

a)  y = -1/(x - 3)

b) y = x/(x - 1)(x + 3)

c) y = (x² - 9)/(x - 3)

d) y = 1/(x² + 5)

e) y = 1/(x + 3)²

f) y = x²/(x + 3)

Solution