### 9.3 Graphing General Rational Functions

```9.3 Graphing General
Rational Functions
The Friedland Method
Rational Function: f(x) = p(x)/q(x)
Let p(x) and q(x) be polynomials with no mutual factors.
p(x) = amxm + am-1xm-1 + ... + a1x + a0
Meaning: p(x) is a polynomial of degree m
Example: 3x2+2x+5; degree = 2
q(x) = bnxn + bn-1xn-1 + ... + b1x + b0
Meaning: q(x) is a polynomial of degree n
Key Characteristics
x-intercepts are the zeros of p(x)
Meaning: Solve the equation: p(x) = 0
Vertical asymptotes occur at zeros of q(x)
Meaning: Solve the equation: q(x) = 0
Horizontal Asymptote depends on the
degree of p(x), which is m, and the degree
of q(x), which is n.
If m < n, then x-axis asymptote (y = 0)
If m = n, divide the leading coefficients
If m > n, then NO horizontal asymptote.
Graphing a Rational Function where m < n
4
Example: Graph y =
2
x +1
State the domain and range.
x-intercepts: None; p(x) = 4 ≠ 0
Vertical Asymptotes: None; q(x) = x2+
1. But if x2+ 1 = 0 ---> x2 = -1. No
real solutions.
Degree p(x) < Degree q(x) -->
Horizontal Asymptote at y = 0 (xaxis)
Let’s look at the picture!
We can see that the domain is ALL REALS
while the range is 0 < y ≤ 4
Graphing a rational function where m = n
2
2
x 3x
Graph y =
4
x-intercepts: 3x2 = 0 ---> x2 = 0 --->
x = 0.
Vertical asymptotes: x2 - 4 = 0
---> (x - 2)(x+2) = 0 ---> x= ±2
Degree of p(x) = degree of q(x) --->
3 ÷ 1 = 3.
Horizontal Asymptote: y = 3
Here’s the picture!
x
y
-4
4
5.
-3
4
-1 -1
0
0
1
-1
5.
4
4
3
4
You’ll notice the three branches.
This often happens with overlapping
horizontal and vertical asymptotes.
The key is to test points in each region!
Graphing a Rational Function where m > n
Graph y =
x2- 2x 3
x+4
x-intercepts: x2- 2x - 3 = 0
---> (x - 3)(x + 1) = 0 ---> x = 3, x = -1
Vertical asymptotes: x + 4 = 0 ---> x = -4
Degree of p(x) > degree of q(x) ---> No
horizontal asymptote
x
y
12 20.6
-9 -19.2
-6
22.5
-2 2.5
0
0.75
2
-0.5
6
Picture time!
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.
You won’t have to worry about that.
The Big Ideas
Always be able to find:
x-intercepts (where numerator = 0)
Vertical asymptotes (where
denominator = 0)
Horizontal asymptotes; depends on
degree of numerator and denominator
Sketch branch in each region
```