POINTS OF DISCONTINUITY Rational Function fx is a

POINTS OF DISCONTINUITY

Rational Function – f(x) is a function that can be written as

Points of Discontinuity a) b) A graph is continuous if no value of x makes the denominator zero, so it has no breaks. A graph is discontinuous if it contains a REAL number which produces zero when substituted into the denominator. A point of discontinuity is a RESTRICTION; where the denominator equals zero because it breaks the graph at that point.

Look at the graph and find where the denominators would be restricted.

Example 1: Finding points of discontinuity. None No REAL root

2 Types of Discontinuity: Vertical Asymptotes(x =) and Holes a. A POD is a vertical asymptote if it does NOT cancel out with any common factor in the numerator. (Draw asymptotes with dashed lines) b. If the POD DOES cancel out with a common factor then there is a hole in the graph. (Holes are a point! Write them (x, y). Note: A vertical asymptote COVERS a hole if they overlap.

Example 2: Finding Vertical Asymptotes and Holes

Example 2: Finding Vertical Asymptotes and Holes

Example of a “hole” in the graph,

Horizontal Asymptotes (y=)

Example 3: Find the HA of each function. None

Example 3 Find the Horizontal Asymptotes None

Homework Page 495 -496 #’s 2 -8 even, 10 -16, 20 -24 even

Horizontal Asymptotes (y=) a. If the degree of the denominator is greater than the degree of the numerator the horizontal asymptote is y = 0 b. If the degree of the numerator is greater than the degree of the denominator the graph has NO horizontal asymptote. c. If the degree of the denominator is equal to the degree of the numerator the horizontal asymptote is