The Friedland Method
9.3 Graphing General Rational Functions
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-1x
n-1 + ... + b1x + b0
Meaning: q(x) is a polynomial of degree n
Example: 7x5-3x2+2x-1; degree = 5
Rational Function: f(x) = p(x)/q(x)
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.
Key Characteristics
Example: Graph y =
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 (x-axis)
Graphing a Rational Function where m < n •4
•x2+1
We can see that the domain is ALL REALS while the range is 0 < y ≤ 4
Let’s look at the picture!
Graph y =
x-intercepts: 3x2 = 0 ---> x2 = 0 ---> x = 0.Vertical asymptotes: x2 - 4 = 0
---> (x - 2)(x+2) = 0 ---> x= ±2Degree of p(x) = degree of q(x) ---> divide the leading coefficients ---> 3 ÷ 1 = 3.
Horizontal Asymptote: y = 3
Graphing a rational function where m = n
•3x2
•x2-4
Here’s the picture!•x •y•-4 •4
•-3•5.4
•-1 •-1•0 •0•1 •-1
•3•5.4
•4 •4You’ll notice the three branches.
This often happens with overlapping horizontal and vertical asymptotes.
The key is to test points in each region!
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 p(x) > degree of q(x) ---> No horizontal asymptote
Graphing a Rational Function where m > n
•x2- 2x - 3•x + 4
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.
Picture time!•x •y•-12
•-20.6
•-9•-19.2
•-6•-22.5
•-2 •2.5
•0•-0.75
•2 •-0.5•6 •2.1
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