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