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EMBED Equation.DSMT4  Example 1: Using the graph of  EMBED Equation.DSMT4  as a guide, describe the transformation(s) and then graph. a.  EMBED Equation.DSMT4   Domain: __________________ Range: ___________________ Vertical Asymptote: _________ Horiz. Asymptote: ___________ b.  EMBED Equation.DSMT4   Domain: __________________ Range: ___________________ Vertical Asymptote: _________ Horiz. Asymptote: ___________ c.  EMBED Equation.DSMT4   Domain: __________________ Range: ___________________ Vertical Asymptote: _________ Horiz. Asymptote: ___________ Rational FunctionsFor a rational function of the form  EMBED Equation.DSMT4 , 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 ____________ Example 2: Identify the asymptotes, domain, and range of the following functions WITHOUT graphing. a.  EMBED Equation.DSMT4  Vertical Asymptote: _________ Horiz. Asymptote: ___________ Domain: __________________ Range: ___________________b.  EMBED Equation.DSMT4  Vertical Asymptote: _________ Horiz. Asymptote: ___________ Domain: __________________ Range: ___________________ 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, exponential, 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, lets focus on graphs that have holes in them. ( Holes in GraphsIf a rational function has the same factor  EMBED Equation.DSMT4  in both the numerator and the denominator, then there is a hole in the graph at the point where  EMBED Equation.DSMT4 , unless the line  EMBED Equation.DSMT4  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? Lets see Example 3: Identify holes in the graph of each function and then graph. a.  EMBED Equation.DSMT4   Hole @  EMBED Equation.DSMT4  Coordinates of the Hole:  EMBED Equation.DSMT4  Domain: __________________ Range: ___________________b.  EMBED Equation.DSMT4   Hole @  EMBED Equation.DSMT4  Coordinates of the Hole:  EMBED Equation.DSMT4  Domain: __________________ Range: ___________________ DAY TWO: Zeros and Vertical Asymptotes of Rational FunctionsIf  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are polynomial functions in standard form with no common factors other than 1, then the function has zeros at each real value of  EMBED Equation.DSMT4  for which  EMBED Equation.DSMT4 . A vertical asymptote at each real value of  EMBED Equation.DSMT4  for which  EMBED Equation.DSMT4 . Example 4: Identify the zeros and the vertical asymptotes (if any) of each rational function. a.  EMBED Equation.DSMT4 b.  EMBED Equation.DSMT4 c.  EMBED Equation.DSMT4  CAREFUL!  Some rational functions, including those whose graphs are hyperbolas, have a horizontal asymptote. The existence 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 except at the right and left extremes. 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 extremes 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, lets just learn how to find the horizontal asymptotes. Horizontal Asymptotes of Rational Functions MEMORIZE THESE RULES!If  EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  are polynomial functions in standard form with no common factors other than 1. The graph of  EMBED Equation.DSMT4  has at most one horizontal asymptote. If degree of  EMBED Equation.DSMT4  > degree of  EMBED Equation.DSMT4 , there is NO horizontal asymptote. If degree of  EMBED Equation.DSMT4  < degree of  EMBED Equation.DSMT4 , the horizontal asymptote is the line  EMBED Equation.DSMT4 . If degree of  EMBED Equation.DSMT4  = degree of  EMBED Equation.DSMT4 , the horizontal asymptote is the line  EMBED Equation.DSMT4 .Example 5: Identify the horizontal asymptote of each rational function. a.  EMBED Equation.DSMT4  b.  EMBED Equation.DSMT4  c.  EMBED Equation.DSMT4  d.  EMBED Equation.DSMT4   Now, time to put it all together and graph some basic rational functions. Example 6: Graph each rational function. a.  EMBED Equation.DSMT4   b.  EMBED Equation.DSMT4    c.  EMBED Equation.DSMT4    d.  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