8.2 Rational Functions and Their Graphs Objectives: Identify and evaluate rational functions. Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph.

A rational expression is the quotient of two polynomials. A rational function is a function defined by a rational expression. Determine whether the function is a rational function: No 1.) f(x) = 2.) Yes

The rational function f(x) = 1/x is undefined when x=0. In general, the domain of a rational function is the set of all real numbers except those numbers which make the denominator equal to zero.

Example 2

Rational functions can have horizontal and vertical asymptotes.

Real numbers for which a rational function is not defined are called excluded values. At an excluded value, a rational function may have a vertical asymptote. If x – a is a factor of the denominator of a rational function but not a factor of its numerator, then x = a is a vertical asymptote of the graph of the function.

Example 3

Example 4

Example 5

Vertical stretch by a factor of 4 Translation 1 unit up Translation 2 units to the right

The graph of a rational function may have a hole in it. For example, can be written as because x - 3 is a factor of both the numerator and the denominator, the graph of f has a hole when x = 3. hole when x = 3 hole when x = 3

If x – b is a factor of the numerator and the denominator of a rational function, then there is a hole in the graph of a function when x = b, unless x = b is a vertical asymptote.

Identify all asymptotes and holes in the graph. Example 6. Let: Identify all asymptotes and holes in the graph. Because x-1 is a factor of both the numerator and denominator, the graph has a hole when x = 1 Because x+2 is a factor of only the denominator, there is a vertical asymptote. x= -1 Because the degree of the numerator equals the degree of the denominator there is a horizontal asymptote as y= -1.

Homework Integrated Algebra II- Section 8.2 Level A Academic Algebra II- Section 8.2 Level B