1. Graphing Rational Functions

2. Example: f (x) = is defined for all real numbers except x = 0.
A rational function is a function of the form f(x) = , where P(x) and Q(x) are polynomials and Q(x) = 0.
Example: f (x) = is defined for all real numbers except x = 0.
f(x) = x
f(x) 2
0.5 1
0.1 10
0.01
100
0.001
1000 x
f(x) -2
-0.5 -1
-0.1
-10
-0.01
-100
-0.001
-1000
As x → 0+, f(x) → +∞.
As x → 0–, f(x) → -∞.
Rational Function

3. The line x = a is a vertical asymptote of the graph of y = f(x),
if and only if f(x) → + ∞ or f(x) → – ∞ as x → a + or as x → a –. x
x = a
as x → a – f(x) → + ∞ x
x = a
as x → a – f(x) → – ∞ x
x = a
as x → a + f(x) → + ∞ x
x = a
as x → a + f(x) → – ∞
Vertical Asymptote

4. Example 1: Vertical Asymptote
Example: Show that the line x = 2 is a vertical asymptote of the graph of f(x) = y x
100
0.5
x = 2 x
f(x)
1.5 16
1.9
400
1.99
40000 2 -
2.01
2.1
2.5
f (x) =
Observe that:
x→2–, f (x) → – ∞
This shows that x = 2 is a vertical asymptote.
x→2+, f (x) → + ∞
Example 1: Vertical Asymptote

5. Example 2: Vertical Asymptote
A rational function may have a vertical asymptote at x = a for any value of a such that Q(a) = 0.
Example: Find the vertical asymptotes of the graph of f(x) =
Example 2: Vertical Asymptote
Set the denominator equal to zero and solve.
Solve the quadratic equation x2 + 4x – 5.
(x – 1)(x + 5) = 0
Therefore, x = 1 and x = -5 are the values of x for which f may have a vertical asymptote.
As x →1– , f(x) → – ∞.
As x → -5–, f(x) → + ∞.
As x →1+, f(x) → + ∞.
As x →-5+, f(x) → – ∞.
x = 1 is a vertical asymptote.
x = -5 is a vertical asymptote.

6. Example 3: Vertical Asymptote
Example: Find the vertical asymptotes of the graph of f(x) =
1. Find the roots of the denominator.
0 = x2 – 4 = (x + 2)(x – 2)
Possible vertical asymptotes are x = -2 and x = +2.
Example 3: Vertical Asymptote
2. Calculate the values approaching -2 and +2 from both sides x → -2, f(x) → -0.25; so x = -2 is not a vertical asymptote.
x → +2–, f(x) → – ∞ and x →+2+, f(x) → + ∞. So, x = 2 is a vertical asymptote. x y
(-2, -0.25)
x = 2
f is undefined at -2
A hole in the graph of f at (-2, -0.25) shows a removable singularity.

7. The line y = b is a horizontal asymptote of the graph of y = f(x)
if and only if f(x) → b + or f(x) → b – as x → + ∞ or as x → – ∞.
Horizontal Asymptote y
y = b
as x → + ∞ f(x) → b – y
y = b
as x → – ∞ f(x) → b – y
y = b
as x → + ∞ f(x) → b + y
y = b
as x → – ∞ f(x) → b +

8. Example 1: Horizontal Asymptote
Example: Show that the line y = 0 is a horizontal asymptote of the graph of the function f(x) = .
As x becomes unbounded positively, f(x) approaches zero from above; therefore, the line y = 0 is a horizontal asymptote of the graph of f.
As f(x) → – ∞, x → 0 –. x y x
f(x) 10
0.1
100
0.01
1000
0.001 –
-10
-0.1
-100
-0.01
-1000
-0.001
f(x) =
y = 0
Example 1: Horizontal Asymptote

9. Example 2: Horizontal Asymptote
Example: Determine the horizontal asymptotes of the graph of f(x) =
Example 2: Horizontal Asymptote
Divide x2 + 1 into x f(x) = 1 –
As x → +∞, → 0– ; so, f(x) = 1 – →1 –. y x
Similarly, as x → – ∞, f(x) →1–.
y = 1
Therefore, the graph of f has y = 1 as a horizontal asymptote.

10. Asymptotes for Rational Functions
Finding Asymptotes for Rational Functions
Given a rational function: f (x) =
P(x) am xm + lower degree terms
Q(x) bn xn + lower degree terms =
Asymptotes for Rational Functions
If c is a real number which is a root of both P(x) and Q(x), then there is a removable singularity at c.
If c is a root of Q(x) but not a root of P(x), then x = c is a vertical asymptote.
If m > n, then there are no horizontal asymptotes.
If m < n, then y = 0 is a horizontal asymptote.
If m = n, then y = am is a horizontal asymptote. bn

11. Horizontal and Vertical Asymptotes
Example: Find all horizontal and vertical asymptotes of f (x) =
Horizontal and Vertical Asymptotes
Factor the numerator and denominator.
The only root of the numerator is x = -1. The roots of the denominator are x = -1 and x = 2 .
x = 2
Since -1 is a common root of both, there is a hole in the graph at -1 . x y
Since 2 is a root of the denominator but not the numerator, x = 2 will be a vertical asymptote.
y = 3
Since the polynomials have the same degree, y = 3 will be a horizontal asymptote.

12. A slant asymptote is an asymptote which is not vertical or horizontal.
Example: Find the slant asymptote for f(x) =
Divide: x y
y = 2x - 5
x = -3
As x → + ∞, → 0+.
As x → – ∞, → 0–.
Therefore as x → ∞, f(x) is more like the line y = 2x – 5.
The slant asymptote is y = 2x – 5.
Slant Asymptote