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.

Take the above rational function, and complete the following charts. XY XY As ‘x’ gets very big, what is the behavior of y? A ‘x’ gets very small what is the behavior of y? XY XY

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?

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.

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

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

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.

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

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)

CHECK USING YOUR CALCULATOR

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