1. 2.6-2.7 Graphing General Rational Functions
The Friedland Method

2. What’s the big idea?
We’re always looking for shortcuts to make sketching easier. The next two questions shed some light on a function’s graph:
What happens to f(x) as x as gets closer and closer to where the denominator is zero?
What happens to f(x) as x goes way right (positive infinity) or way left (negative infinity)?
The answer to the first question deals with asymptotes. The answer to the second question deals with the end behavior of a function

3. Talkin’ bout Asymptotes.
Let’s look at f(x) = 1/(x-2)
We know that x ≠ 2. But what happens as x gets close to 2?
If x = , f(x) = -10,000
If x = , f(x) = 10,000.
As x approaches the asymptote from the left, the function goes to -∞
And as x approaches the asymptote from the right, the function goes to ∞
We write that as:
x  2- f(x)  -∞
x  2+ f(x)  ∞

4. Vertical Asymptotes
The line x = a is a vertical asymptote of f(x) if f(x)  ∞ or f(x)  -∞ as x approaches a from the left or right side.
NOTE: We are not guaranteed a vertical asymptote simply because we have a restricted domain value.
Example: f(x) = 3x/x
Notice how the x just cancels out?
When that happens, you get a hole instead of a vertical asymptote.
So the equation becomes y = 3 but with a hole at x = 0 (the domain restriction)

5. Rational Function: f(x) = N(x)/D(x)
Let N(x) and D(x) be polynomials with no mutual factors.
N(x) = anxn + an-1xn a1x + a0
Meaning: N(x) is a polynomial of degree n
Example: 3x2+2x+5; degree = 2
D(x) = bmxm + bm-1xm b1x + b0
Meaning: D(x) is a polynomial of degree m
Example: 7x5-3x2+2x-1; degree = 5

6. Key Characteristics x-intercepts are the zeros of N(x) (the top)
Meaning: Solve the equation: N(x) = 0
Vertical asymptotes occur at zeros of D(x)
Meaning: Solve the equation: D(x) = 0 (bottom)
Horizontal Asymptote depends on the degree of N(x), which is n, and the degree of D(x), which is m.
If n < m (“bottom heavy”), then y = 0
If m = n (tie), divide the leading coefficients
If m > n (“top heavy”), then NO horizontal asymptote, do long division.

7. Graphing a rational function where m = n
3x2
x2-4
Graph y =
x-intercepts: Set top = 0 and solve!
3x2 = 0  x2 = 0  x = 0.
Vertical Asymptotes: Set bottom = 0 and solve
x2 - 4 = 0  (x - 2)(x+2) = 0  x= ±2
Degree of N(x) = degree of D(x)  top and bottom tie  divide the leading coefficients
Y = 3/1 = 3.
Horizontal Asymptote: y = 3

8. You’ll notice the three branches.
Here’s the picture! x y -4 4 -3
5.4 -1 1 3
You’ll notice the three branches.
This often happens with overlapping horizontal and vertical asymptotes.
The key is to test points in each region!
Domain:
x ≠ ±2
Range:

9. Graphing a Rational Function where m < n
Example: Graph y =
State the domain and range.
x-intercepts: None; N(x) = 4 and 4 ≠ 0
Vertical Asymptotes: None; D(x) = x2 + 1.
But if x2 + 1 = 0  x2 = -1. No real solutions.
Degree N(x) < Degree D(x) --> Horizontal Asymptote at y = 0 (x-axis) 4
x2 + 1

10. Let’s look at the picture!
Domain:
Range:

11. Graphing a Rational Function where n > m
Graph y =
x-intercepts: x2- 2x - 3 = 0
(x - 3)(x + 1) = 0  x = 3, x = -1
Vertical asymptotes: x + 4 = 0  x = -4
Degree of N(x) > degree of D(x)
No horizontal asymptote, do long division.
x2- 2x - 3
x + 4

12. Picture time! x y -12 -20.6 -9 -19.2 -6 -22.5 -2 2.5 -0.75 2 -0.5 6
-0.75 2
-0.5 6
2.1
Not a lot of pretty points on this one.
This graph actually has a special type of asymptote called “oblique.” It’s drawn in purple. Don’t worry, I’ll show you how it works.

13. Oblique Asymptotes Equation of Asymptote! Remainder is Irrelevant
These happen when the top polynomial overpowers the bottom polynomial.
“Top Heavy” means N(x) has a higher degree.
The equation of the oblique asymptote is just the result when you long divide the two polynomials.
Prior example:
Remainder is
Irrelevant

14. The Big Ideas x-intercepts (where numerator = 0)
Always be able to find:
x-intercepts (where numerator = 0)
Vertical asymptotes (where denominator = 0)
Horizontal asymptotes:
If bottom heavy: y=0 asymptote.
If they tie: divide leading coefficient.
If top heavy: Long divide.
Plot points in each region

15. Practice Problems 6 x2 + 3 No x-intercepts No vertical asymptote
Graph y = 6
x2 + 3 x y 2 -1
1.5 1 -2
6/7
No x-intercepts
No vertical asymptote
H.A.: y = 0

16. Practice Problems x2 - 4 x-int: x2 – 4 = 0; x = ±2
Graph y =
x2 - 4
x + 1
x-int: x2 – 4 = 0; x = ±2
V.A.: x + 1 = 0  x = -1
H.A.: Top wins:
No H.A.
Oblique: y = x - 1 x y -4 -3
-5/2

17. Practice Problems 2x2 x-int: 2x2 = 0  x = 0 x2 - 1
Graph y =
x-int: 2x2 = 0  x = 0
V.A.: x2 – 1 = 0  x = ±1
H.A.: Tie 
y = 2/1 = 2 x y 2
8/3 3
9/4 -2 -3