1. 2.6 (Day One) Rational Functions & Their Graphs Objectives for 2.6 –Find domain of rational functions. –Identify vertical asymptotes. –Identify horizontal asymptotes. –Use transformations to graph rational functions. –Graph rational functions. –Identify slant (oblique) asymptotes. –Solve applied problems with rational functions. Pg. 342 #2-8 (evens), 22-36(evens), 72a, 74a, 76a, 78a

2. Find the domain of each rational function. 1.f(x) = 2.f(x) = 3.f(x) =

3. Vertical asymptotes A vertical asymptote is a vertical line that marks a single value of x excluded from the domain of the function. Look for domain restrictions. If there are values of x which result in a zero denominator, these values would create either a hole in the graph or a vertical asymptote line. If the factor that creates a zero denominator cancels with a factor in the numerator, there is a hole. Example: Hole at x=-5 If you cannot cancel the factor from the denominator, a vertical asymptote line exists. Example: Vertical line asymptotes at x=3 and -3 An asymptote is a line (or curve) that forms a boundary that a function approaches, moving closer and closer; sometimes over infinite regions.

4. Find the vertical asymptote(s) or hole, if any, in the graph of each rational function. 4. 5. 6. 7.

5. What is the end behavior of this rational function? Meaning, what is the graph dong on the far left and far right ? If you are interested in the end behavior, you are concerned with extremely large and extremely small values of x. As x approaches positive or negative infinity, the highest degree term becomes the only term of interest. The other terms become negligible (of very little importance) in comparison. So, only examine the ratio of the highest degree term in the numerator over the highest degree term of the denominator (ignore all others!) As x gets large, becomes THEREFORE, a horizontal asymptote exists, y=3

6. For the graph of If n<m (the numerator is of lesser degree than the denominator), then the end behavior is limited by the horizontal asymptote If n=m (the numerator is of equal degree with the denominator), then the end behavior limited by the horizontal asymptote If n (the numerator) is exactly one degree higher than m (the denominator), then the end behavior is limited by a slanted line asymptote. If n (the numerator) is more than one degree higher than m (the denominator), then the end behavior is limited by an asymptote that can be a curve. NOTE: The asymptotic curve is NOT covered in this precalculus textbook.

7. Find the horizontal or the slant asymptote, if any asymptote exists, of the graph of each of the following rational functions. 8. 9.

8. 10. 11.

9. Graph of this rational function It has a vertical asymptote at x=3 and a slant asymptote at y = 4x+(21/2)