1. Rational Functions Lesson Goals -recognize asymptotic presence -determine and locate vertical asymptotes -determine and locate horizontal asymptotes -graph rational functions -distinguish the difference between rational functions and functions previously encountered. Standards Covered -Describe and compare the characteristics of the following families of functions: quadratics with complex roots, polynomials of any degree, logarithms, and rational functions; e.g., general shape, number of roots, domain and range, asymptotic behavior.

2. Take the above rational function, and complete the following charts. XY 1 10 100 1000 10000 XY 1..1.01.001.0001 As ‘x’ gets very big, what is the behavior of y? A ‘x’ gets very small what is the behavior of y? XY -10 -100 -1000 -10000 XY -.1 -.01 -.001 -.0001

3. TI-83/84 y 1 = 1/x go to table and confirm your assumptions. go to graph, does the graph ever leave the 1 st quadrant? what happens when ‘x’ is negative?

5. f(x) = 1/x is confined to the 1 st and 3 rd quadrants by what are called asymptotes. An asymptote is essentially, a line, that a graph approaches but never crosses. The line that a curve approaches as you follow the curve to infinity. A graph is undefined at an asymptote.

6. Where does y = 1/x blow up? Thinking about domains, it should make sense that ‘x’ cannot ever equal zero, because anything in the form a/0 is undefined. If you let y = 0, then solve 0 = 1/x this blows up as well. So, what does this mean?  vertical asymptote at x=0  horizontal asymptote at y=0

7. Vertical Asymptotes Found by factoring the denominator of a rational function, and locating the zeros of each factor. +++graphing+++ after finding VA, see if y  (+) infinity see if y  (-) infinity

8. Horizontal Asymptote (if present) divide both the numerator and denominator by the highest power of x that appears in the denominator. Horizontal asymptotes are different than vertical asymptotes. -Horizontal asymptotes only behave as asymptotes in the event of determining end behavior of a graph. (as x gets really big or really small) -The difference is that horizontal asymptotes can be crossed, and are usually crossed near the origin.

9. HA cont. a)If n < m, then HA at y=0, because bx m gets very big...a n /very big # =0 b)If n=m, then HA at y = a n /b m c)If n>m, then no horizontal asymptote…check for slant asymptote

10. Put it all together 1.)factor 2.)cancel like terms (holes appear at the cancelled terms) POINT DISCONTINUITY 3.)find VA by setting denominator = 0 4.)find HA by dividing leading terms (non-factored) 5.)find x-intercepts. (when y=0) 6.)find y-intercept (when x=0) 7.)graph (find intermediate points if needed)

13. CHECK USING YOUR CALCULATOR

18. Homework Pg. 322 10, 11, 12, 14, 25, 31 Tomorrow  slant/oblique asymptotes, holes, asymptotes and technology