1. REACH ASSESSMENT REVIEW SKETCHING Y Intercept (Monday) Zeroes (Monday) Asymptotes (Wednesday) THURSDAY QUIZ (YTD + KA TRIGONOMETRY) Week 3 –Day #s 8-12 (week of Reach, Explore, Plan)

2. VERTICAL/HORIZONTAL/SLANT MR. POULAKOS Asymptotes

3. Asymptotes of Rational Functions, A Rational Function is: An Asymptote is, essentially, a line that a function approaches, but never touches or crosses. There are three types:  Vertical Asymptote  Horizontal Asymptote  Slant Asymptote (future lesson) The Asymptote is represented on x-y coordinate system as a dashed line “- - - - - - - ”  Why? - - - - - - - - - - - - - - - - - horizontal Vertical

4. 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 zero, 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

5. 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

6. REMEMBER:  ASYMPTOTES ARE ALWAYS LINES.  THEY ARE LINES THAT A GRAPH APPROACHES BUT DOES NOT TOUCH OR CROSS (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 … Vertical asymptotes – Sample Problems x = 5 x = +4 and x = –4 x = – 4 and x = – 2

9. Rational FunctionVertical Asymptote (s) is/are at … Vertical asymptotes – Sample Problems x = 3 x = +5 and x = –5 x = – 4 and x = 6

10. 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 …

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

12. Slant asymptotes Finding the Slant asymptote(s) is similar to Horizontal in that we look at the degrees. Compare the degree of the numerator (n) to that of the denominator (m).  If n=m+1 (exactly), then there is a slant asymptote and it is determined by:  Performing long division  dividend ÷ divisor.  The slant asymptote is the quotient without including the remainder. y=“quotient” f(x)= (x 2 –x−2) ÷ (x−1)  slant: y=x f(x)= (x 2 +4x+13) ÷ (x+1)  slant: y=x+3 

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14. Asymptotes: Summary Note: n is the degree of the numerator & m that of the denominator

15. Sketching To Sketch Rational Functions  hints are in [ ]s: 1) Simplify, if possible. 2) Find & plot y intercepts [x=0] 3) Find & plot the x intercepts [N(x)=0]. 4) Find & sketch any vertical asymptotes [D(x)=0] 5) Find & sketch any horizontal asymptotes [“degrees”] 6) Find & sketch any slant asymptotes [n=m+1] 7) Plot at least one point between and one point beyond each x-intercept and vertical asymptote. 8) Use smooth curves to complete the graph between and beyond the vertical asymptotes.