Graphing General Rational Functions

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

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 = 1.9999, f(x) = -10,000

If x = 2.0001, f(x) = 10,000.

-3 8

-4

4

x

y

Ah! 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) ∞

Vertical Asymptotes

-3 8

-3

8

x

y

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) (the top)

Meaning: Solve the equation: p(x) = 0

Vertical asymptotes occur at zeros of q(x)

Meaning: Solve the equation: q(x) = 0 (bottom)

Horizontal Asymptote depends on the degree of p(x), which is m, and the degree of q(x), which is n.

If m < n (bottom wins), then y = 0

If m = n (tie), divide the leading coefficients

If m > n (top wins), then NO horizontal asymptote

Key Characteristics

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 p(x) = degree of q(x) top and bottom tie divide the leading coefficients

Y = 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 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!

Domain: x ≠ ±2Range: y > 3 & y ≤ 0

Example: Graph y =

State the domain and range.

x-intercepts: None; p(x) = 4 and 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: 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 - 3x + 4

Not a lot of pretty points on this one. This graph actually has a special type of asymptote called “slant.” It’s drawn in purple. Don’t worry, I’ll show you how it works.

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

Slant AsymptotesThese happen when the top polynomial overpowers the bottom polynomial.

“Overpowers” means has a higher degree.

The equation of the slant asymptote is just the result when you divide the two polymomials out.

Prior example:

Equation of Asymptote!

Remainder is

Irrelevant

The Big IdeasAlways be able to find:

x-intercepts (where numerator = 0)

Vertical asymptotes (where denominator = 0)

Horizontal asymptotes:

If bottom wins: x-axis asymptote

If they tie: divide leading coeff.

If top wins: No horizontal asymp.

Sketch branch in each region (plot points)

Practice ProblemsGraph y =

6

x2 + 3

No x-intercepts No vertical asymptoteH.A.: y = 0

x y

0 2

-1 1.5

1 1.5

-2 6/7

2 6/7

Practice ProblemsGraph y =

x2 - 4

x + 1

x y

0 -4

-3 -5/2

x-int: x2 – 4 = 0; x = ±2V.A.: x + 1 = 0 x = -1

H.A.: Top wins: No H.A.Slant: y = x - 1

Practice ProblemsGraph y =

2x2

x2 - 1 x-int: 2x2 = 0 x = 0V.A.: x2 – 1 = 0 x = ±1

H.A.: Tie y = 2/1 = 2

x y

2 8/3

3 9/4

-2 8/3

-3 9/4

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