1. VERTICAL AND HORIZONTAL (TUESDAY) (WEDNESDAY/THURS.) COLLEGE ALGEBRA MR. POULAKOS MARCH 2011 Asymptotes

2. Asymptotes of Rational Functions, A Rational Function is: An Asymptote is, essentially, a line that a graph approaches, but does not touch or cross. There are two types:  Vertical Asymptote  Horizontal Asymptote The Asymptote is represented on x-y coordinate system as a dashed line “- - - - - - - ”  Why? - - - - - - - - - - - - - - - - - horizontal Vertical

3. Vertical asymptotes The Vertical asymptote is a line that the graph approaches but does not intersect. Vertical asymptotes almost always occur because the denominator of a fraction has gone to 0, but the numerator (top) has not. For example, Note that as the graph approaches x=2. From the left, the curve drops rapidly towards negative infinity. This is because the numerator is staying at 4, and the denominator is getting close to 0.numerator

4. Horizontal Asymptote The Horizontal asymptote is also a line that the graph approaches but does not intersect In the following graph of y=1/x, the line approaches the x- axis (y=0) as x gets larger. But it never touches the x-axis. No matter how far we go into infinity, the line will not actually reach y=0, but it will keep getting closer and closer. This means that the line y=0 is a horizontal asymptote. The domain for y=1/x is all real numbers except 0

5. Horizontal asymptotes Horizontal asymptotes occur most often when the function is a fraction where the top remains positive, but the bottom goes to infinity. Going back to the previous example, y=1/x is a fraction. When we go out to infinity on the x-axis, the top of the fraction remains 1, but the bottom gets bigger and bigger. As a result, the entire fraction actually gets smaller, although it will not hit zero. The function will be 1/2, then 1/3, then 1/10, even 1/10000, but never quite 0. Thus, y=0 is a horizontal asymptote for the function y=1/x

6. REMEMBER:  ASYMPTOTES ARE ALWAYS LINES.  THEY ARE LINES THAT A GRAPH APPROACHES BUT DOES NOT TOUCH (DOES NOT INTERSECT) Finding Asymptotes

7. Vertical asymptotes Remember, Vertical asymptotes almost always occur because the denominator of a fraction has gone to 0, but the top has not  the zeroes of the denominator. Therefore, set the denominator to zero and solve for the variable. For example, x–7=0 x=7 is the asymptote. Factor : x 2 –16=0  (x–4)(x+4) = 0 Solve: a) x–4=0 and b) x+4=0, Therefore, there are 2 asymptotes. a) x = +4 and b) x = –4

8. Rational FunctionVertical Asymptote (s) is/are at … x = 5 x = +4 and x = –4 x = – 4 and x = – 2 Vertical asymptotes – Sample Problems Page

9. Horizontal asymptotes Finding the Horizontal asymptote(s) are more challenging… Compare the degree of the numerator (n) to that of the denominator (m).  If n<m, then the horizontal asymptote is at y = 0.  If n=m (the degrees are the same), then the asymptote is at y = 1 st coefficient of numerator ÷ 1 st coefficient of denominator  If n>m, then there are no Horizontal asymptotes. Examples follow … See page 338

10. Examples -- Horizontal Asymptotes n < m Asymptote is at y=0 n = m Asymptote is at y=a n /b m Asymptote is at… n >m No Asymptote y=6/3= 2 y=2/5 y=6/4= 3/2