RATIONAL FUNCTIONS AND ASYMPTOTES

Asymptotes are lines that the curve approaches at the edges of the coordinate plane. 

Types of asymptotes :

• Horizontal asymptote
• Vertical asymptote
• Oblique or slant asymptote

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

Problem 1 :

y = 2/(x – 6)

Solution :

 y = 2/(x – 6)

Degree of numerator = 0

Degree of denominator = 1

degree of numerator < degree of denominator

So, equation of the horizontal asymptote is y = 0 which is the x – axis.

Problem 2 :

y = (x + 2)/(x – 4)

Solution :

y = (x + 2)/(x – 4)

Degree of numerator = 1

Degree of denominator = 1

degree of numerator = degree of denominator

y = leading coefficient of N(x)/leading coefficient of D(x).

So, equation of the horizontal asymptote is y = 1.

More problems on horizontal asymptotes

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

Problem 3 :

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

Solution :

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

We see x – 2 as a common factor in both numerator and denominator.

After cancel x – 2. We get,

1/(x + 2)

To find vertical asymptote, we equate the denominator to 0.

x + 2 = 0

x = -2

We see the hole at x = 2 in the graph.

x – 2 = 0 we will get hole at x = 2.

So, the vertical asymptote at x = -2; hole at x = 2.

More problems on vertical asymptotes

Find the oblique asymptote of the rational function

Problem 4 :

f(x) = (x² + 8x – 20)/(x – 1)

Solution :

f(x) = (x² + 8x – 20)/(x – 1)

y = (x + 9) -11/(x – 1)

So, the oblique asymptote of the rational function is

y = x + 9

More problems on oblique asymptotes

Problem 5 :

Consider the function f(x) = (4x - 3) / (x + 1)

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 behaviors of the function near its horizontal asymptote.

e) Determine the domain and range.

f ) Determine the positive and negative intervals.

g) Sketch the graph

Solution :

f(x) = (4x - 3) / (x + 1)

a) The vertical asymptote is at x = -1

b) The intervals are (-∞, -1) and (-1, ∞)

• When x ∈ (-∞, -1), f(x) will be positive. That is, when x < -1, f(x) is positive

• When x ∈ (-1, 3/4), f (x) will be negative, when -1 < x < 3/4, f(x) is negative
• When x ∈ (3/4,∞), f(x) will be positive. That is, when x > 3/4, f(x) is positive.

y-intercept is -3/2.

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

Equation of horizontal asymptote is y = 4/1

y = 4

d) End behavior :

• x --> -∞ then f(x) --> 4
• x --> ∞ then f(x) --> 4

e) Domain is all real numbers except x = -1

Range is all real values except y = 4

f) 

• x ∈ (-∞, -1), f(x) will be positive.
• x ∈ (-1, 3/4), f (x) will be negative
•  x ∈ (3/4,∞), f(x) will be positive

Problem 6 :

Read each set of conditions. State the equation of a rational function of the form

f(x) = (ax + b) / (cx + d)

that meets these conditions and sketch the graph.

• Vertical asymptote at x = -2, horizontal asymptote at y = 0
• negative when x ∈ (-∞, -2), positive when x ∈ (-2, ∞) always decreasing.

Solution :

Since the vertical asymptote is x = -2, then (x + 2) is the denominator. Horizontal asymptote is y = 0 then highest exponent of the numerator should be lesser than the highest exponent of the denominator.

So, the required function may be 

f(x) = 1/(x + 2)

Some more functions also can be created.