Algebra 2 Notes Name: Section 8.4 Rational Functions. A function is a function whose rule can be written as a of. 1 x. =. Its graph is a, f x

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1 Algebra Notes Name: Section 8. Rational Functions DAY ONE: A function is a function whose rule can be written as a of two polynomials. The parent rational function is f. Its graph is a, which has two separate branches. You will learn more about hyperbolas in Chapter 0. Like logarithmic and eponential functions, rational functions may have. The function f. has a vertical asymptote at and a horizontal asymptote at y Parent Rational Function Domain: Range: f : Vertical Asymptote: Horizontal Asymptote: y -intercept: Zero(s) or -intercept(s): The rational function f other types of functions. can be transformed by using methods similar to those used to transform f a + h k

2 Eample : Using the graph of f a. g h + b. as a guide, describe the transformation(s) and then graph. c. g + Rational Functions For a rational function of the form f a + k h - the graph is a hyperbola - there is a vertical asymptote at the line, and the domain is, - there is a horizontal asymptote at the line, and the range is Eample : Identify the asymptotes, domain, and range of the following functions WITHOUT graphing. a. g + + b. g 5

3 A function is a function whose graph has one or more gaps or breaks. Many rational functions are discontinuous functions. A function is a function whose graph has no gaps or breaks. The functions you have studied before this, including linear, quadratic, polynomial, eponential, and logarithmic functions, are continuous functions. Tomorrow we will get into some of the more challenging rational functions and how to graph them, but for today, let s focus on graphs that have holes in them. Holes in Graphs If a rational function has the same factor b in both the numerator and the denominator, then there is a hole in the graph at the point where b, unless the line b is a vertical asymptote. WARNING: Your calculator graph is not very good about showing you the holes, when present. And your table will not give you the coordinates of the hole. You will need to be able to find the hole on your won. How, you ask? Let s see Eample : a. f + Identify holes in the graph of each function and then graph. b. g + Coordinates of the Hole:, Coordinates of the Hole:,

4 DAY TWO: Zeros and Vertical Asymptotes of Rational Functions If f p, where p and q are polynomial functions in standard form with no common factors other q than, then the function has p. - zeros at each real value of for which 0 - A vertical asymptote at each real value of for which q( ) 0. Eample : a. f Identify the zeros and the vertical asymptotes (if any) of each rational function. b. g + 5+ c. h + CAREFUL! Some rational functions, including those whose graphs are hyperbolas, have a horizontal asymptote. The eistence and location of a horizontal asymptote depends on the of the polynomials that make up the rational function. Big Time Warning: A horizontal asymptote has NO effect on the graph ecept at the right and left etremes. In particular, it is NOT TRUE that a graph can never cross its horizontal asymptote. The graph might cross its horizontal asymptote many times toward the middle of the graph, or in other words, close to its vertical asymptotes. Only when the graph makes its final approach to the asymptote, at the left or right etremes of the graph, will it squeeze up against the asymptote without ever touching it. (By the way, it IS TRUE that the graph can never touch one of its vertical asymptotes.) Before we worry about graphing rational functions with horizontal asymptotes, though, let s just learn how to find the horizontal asymptotes. Horizontal Asymptotes of Rational Functions MEMORIZE THESE RULES! If f p, where p and q are polynomial functions in standard form with no common factors other q than. The graph of f has at most one horizontal asymptote. - If degree of p > degree of q, there is NO horizontal asymptote. - If degree of p < degree of q, the horizontal asymptote is the line y 0. - If degree of p degree of q, the horizontal asymptote is the line y leading coefficient of leading coefficient of p q.

5 Eample 5: Identify the horizontal asymptote of each rational function. a. f + 5 b. h + 5 c. g f d. + + Now, time to put it all together and graph some basic rational functions. Eample 6: Graph each rational function. a. f + g b. c. h d. f