2. Aim: What are the rational function and asymptotes? Do Now: Graph xy = 4 and determine the domain

3. A rational function is defined as any function that is in the form of p(x)/q(x) where p(x) and q(x) are polynomials and q(x) is not 0. To graph xy = 4, we first need to rewrite xy = 4 into that is a rational function Note: q(x) can equal 1 so any polynomial function is actually a rational function; however we usually use the term only when the function is in the form of a fraction.

4. x can not be 0, therefore, the asymptote of xy = 4 is the line x = 0 (the y-axis) x = 0 is the vertical asymptote, What is the vertical asymptote of To find the vertical asymptote, we simplify let the denominator equal to zero then solve for x. Therefore, the vertical asymptote of f(x) is x = –1

5. For the functionsand Is there any horizontal asymptotes? The horizontal asymptote of y = 8/x is y = 0 (again through observation and experience). The horizontal asymptote of f(x) = 2x + 1 appears to be y = 2. x + 1 Is there any other way we can figure out the horizontal asymptote of any rational functions besides looking at the graphs?

6. There are some simple rules for finding the horizontal asymptote: To do this let’s say the degree of p(x) = n and degree of q(x) = m. 1. If n < m, then the x-axis (y = 0) is the horizontal asymptote. 2. If n = m, then the horizontal asymptote is y = lead. coeff of p(x) / lead. coeff of q(x). 3. If n > m, then there is NO horizontal asymptote.

7. If the degree of numerator is exactly one degree higher than the denominator, then there is a slant asymptote How do we find the equation of the slant asymptote? To find the equation of a slant asymptote, use long division and the quotient is the equation of a slant asymptote Therefore, the slant asymptote is y = x – 2

8. Drill: State the domain of each rational function, then use your calculator and the rules to find the vertical and horizontal asymptotes; if either does not exist say so 1) f(x) = 4 x 2 +1 2) f(x) = 3x 3 + 7x 2 + 2 -4x 3 + 5 3) f(x) = x + 10 |x| + 2 4) f(x) = 2 + x 2 - x