Sketching Rational Functions

all about their asymptotic behaviour

Capturing the Asymptote by flickr user mindtrip

http://webct.merlin.mb.ca/demo

http://mathway.com/

Factor the polynomial completely. Sketch the graph.ƒ(x) = x + 5x + 2x - 8 3 2

Step 4:

Step 3:

Step 2:

Step 1:

Sketch the graph of

Graphing Rational Functions

where a(x) and b(x) are polynomial functions.

Examples

Functions of the form

Graphing Rational Functions

Appearance

Where n is even, the graph looks like this:

Where n is odd, the graph looks like this:

Graphing Rational Functions

Sketching (7 steps)

Step 1: Find the y-intercept (let x = 0)

Step 4: Find the vertical asymptotes by finding the roots of the denominator b(x).

Step 3: Find the roots of the function by finding the roots of the numerator a(x).

Step 2: Factor everything. (Use rational roots theorem if necessary.)

Graphing Rational FunctionsStep 5: Find the horizontal asymptotes by dividing each term in the function by the highest power of x, and take the limit as x goes to infinity. (Use the UNfactored form.)

You will find that, in general, there are three possible results:

i When [degree of numerator < degree of denominator] the horizontal asymptote is y = 0.

iii When [degree of numerator > degree of denominator] there is no horizontal asymptote; however there may be a slant asymptote or a hole in the graph.

ii When [degree of numerator = degree of denominator] the H.A. is the ratio leading coefficient of a(x)

leading coefficient of b(x)

Graphing Rational Functions

Sketching (7 steps)

Step 6: Determine the sign of the function over the intervals defined by the roots and vertical asymptotes. (Use the factored form.)

Step 7: Sketch the graph.

Graphing Rational Functions

Sketching: Example 1 of 4

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

Step 7:

Step 5: