Rational Functions and Their Graphs Objective: Identify and evaluate rational functions and graph them.

Rational Functions A rational expression is the quotient of two polynomials. A rational function is a function defined by a rational expression.

Rational Functions A rational expression is the quotient of two polynomials. A rational function is a function defined by a rational expression. Explain why the following are not rational functions.

Rational Functions A rational expression is the quotient of two polynomials. A rational function is a function defined by a rational expression. Explain why the following are not rational functions. The numerator of the first is not a polynomial, it is an exponential. The denominator of the second is not a polynomial, it is an absolute value.

Definitions Define the following terms: Domain Range Asymptote

Definitions Define the following terms: Domain- All values of x that are defined by the function. A zero in the denominator or a negative square root are bad values. Range-All values of y that are defined the function. Asymptote-Values that are undefined by the function. A zero in the denominator (and not in the numerator) of a fraction is an asymptote.

Domain The domain of a function is all x-values that we are allowed to put into the function; in other words, values that make sense. When looking at domain, we are concerned about two things. 1.Negative square root. 2.Zero in the denominator of a fraction. We will be concerned with the 2 nd constraint.

Example 2

Try This Find the domain of

Try This Find the domain of The domain is all values that don’t make the denominator zero.

Excluded Values 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. The necessary conditions for a vertical asymptote are as follows:

Excluded Values 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. The necessary conditions for a vertical asymptote are as follows:

Example 3

Try This Identify all vertical asymptotes of the graph of

Try This Identify all vertical asymptotes of the graph of The vertical asymptotes are the zeros of the denominator.

Horizontal Asymptotes

I will ask you tomorrow to define/give the conditions where a horizontal asymptote exists. Please look over this definition and be able to explain it.

Example 4

Try This Let. Find the equations of all vertical asymptotes and the horizontal asymptote of the graph of R.

Try This Let. Find the equations of all vertical asymptotes and the horizontal asymptote of the graph of R. The zeros of the denominator are

Try This Let. Find the equations of all vertical asymptotes and the horizontal asymptote of the graph of R. The zeros of the denominator are Since the numerator and denominator are the same power of x, the horizontal asymptote is

Example 5

Holes in Graphs

Example 6

Try This Let. Identify all asymptotes and holes in the graph.

Try This Let. Identify all asymptotes and holes in the graph. Factor the numerator and denominator. The value x = 1 is a zero of the denominator only, so x = 1 is a vertical asymptote. (no horizontal asymptote) The value x = -3 is a zero of both, so x = -3 is a hole in the graph.

Definitions You will need to be able to define the following terms: Domain Range Vertical Asymptote Horizontal Asymptote Hole in the graph

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