1. 2-6 rational functions

2.  Lines l and m are perpendicular lines that intersect at the origin. If line l passes through the point (2,-1), then line m must pass through which of the following points?  A)(0,2)  B)(1,3)  C)(2,1)  D)(3,6)  E)(4,0)

3.  Find the domains of rational functions  Find vertical and horizontal asymptotes of graphs  Use rational functions to model and solve real-life problems

4.  rational function is defined as the quotient of two polynomial functions.  f(x) = P(x) / Q(x)  Here are some examples of rational functions:  g(x) = (x 2 + 1) / (x - 1)  h(x) = (2x + 1) / (x + 3)

6.  An asymptote is a line that the graph of a function approaches but never reaches.

7.  There are two main types of asymptotes: Horizontal and Vertical.

8.  What is vertical asymptote and horizontal asymptote? Here are the general definitions of the two asymptotes. 1.The line is a vertical asymptote if the graph increases or decreases without bound on one or both sides of the line as x moves in closer and closer to. 2.The line is a horizontal asymptote if the graph approaches as x increases or decreases without bound. Note that it doesn’t have to approach as x BOTH increases and decreases. It only needs to approach it on one side in order for it to be a horizontal asymptote.

9.  Vertical Asymptotes of Rational Functions  To find a vertical asymptote, set the denominator equal to 0 and solve for x. If this value, a, is not a removable discontinuity, then x=a is a vertical asymptote.removable discontinuity

10.  1. To find a function's horizontal asymptotes, there are 3 situations.  a. The degree of the numerator is higher than the degree of the denominator.  1. If this is the case, then there are no horizontal asymptotes.  b. The degree of the numerator is less than the degree of the denominator.  1. If this is the case, then the horizontal asymptote is y=0.

11.  The degree of the numerator is the same as the degree of the denominator.  1. If this is the case, then the horizontal asymptote is y = a / d where a is the coefficient in front of the highest degree in the numerator and d is the coefficient in front of the highest degree in the denominator.

13.  In general, the procedure for asymptotes is the following:  set the denominator equal to zero and solve  the zeroes (if any) are the vertical asymptotes  everything else is the domain  compare the degrees of the numerator and the denominator  if the degrees are the same, then you have a horizontal asymptote at y = (numerator's leading coefficient) / (denominator's leading coefficient)  if the denominator's degree is greater (by any margin), then you have a horizontal asymptote at y = 0 (the x-axis)  if the numerator's degree is greater (by a margin of 1), then you have a slant asymptote which you will find by doing long division

14.  The graph has a vertical asymptote at x=_____.  The Equation has horizontal asymptote of  Y=____

15.  Find the domain and all asymptotes of the following function: Then the full answer is: domain: vertical asymptotes: x = ± 3 / 2 horizontal asymptote: y = 1 / 4

16.  Find the domain and all asymptotes of the following function:  domain: all x vertical asymptotes: none horizontal asymptote: y = 0 (the x -axis)

17. Special Case with a "Hole" Find the domain and all asymptotes of the following function:  domain: vertical asymptote: x=2  Horizontal asymptote: None

18.  Do problems 1 -4 on the worksheet

19.  Do problems 17-20 and 25-28 from your book page 148

20.  Today we learned about finding domain and range.  We also learned how to find the vertical and horizontal asymptotes.  Next class we are going to learned about graphs of rational functions