1. Rational Functions

16. To sketch the graph of a rational function: Determine if the function points of discontinuity for the denominator and if they are holes or vertical asymptotes. Sketch in any vertical asymptotes. Determine if the function has a horizontal asymptote. As x gets larger (positive or negative) the graph will approach this line. Calculate values of y for x values that are near the asymptotes. Plot these points and sketch the graph.

33. When a rational equation has a sum or difference of two rational expressions, you can use the LCD to simplify.

35.  Homework: page 532 (1-21) odd  Chapter 9 test Tuesday 4/9 or Wednesday 4/10

36.  Direct and Inverse variation:  If y/x is always equal to the same number, then x and y represent a direct variation.  y=kx  If xy is always the same value then x and y vary inversely  y = k/x

37.  Discontinuities  In rational functions discontinuities occur where values of the variable make the denominator equal to zero.  If this value makes the numerator zero there will be a hole in the graph.  If the value does not make the numerator zero there will be a vertical asymptote in the graph.

38.  Horizontal Asymptotes  Horizontal Asymptotes describe end behavior of graph.  Determined by the degree of the functions in the numerator and denominator.  If degree in denominator is higher, horizontal asymptote at y=0 (X axis)  If degree in numerator is higher there is no horizontal asymptote  If degree is the same, horizontal asymptote occurs at the ratio of the leading coefficients of the numerator and denominator.

39.  Simplify Rational Expressions  Factor all parts of rational expression completely.  Cancel factors that appear in both numerator and denominator.  To multiply: factor and simplify before multiplying.  To divide: Factor, flip second function, simplify and multiply.

40.  Adding or Subtracting Rational Expressions  Must find a common denominator before you can add or subtract.  Complex Rationals:  Multiply top and bottom of rational expression by the Least Common Multiple of all complex denominators

41.  Solving Rational Equations  If possible cross multiply to solve equations.  Determine Least Common Multiple of all rationals and multiply all terms by the LCM.  Always check all of your solutions.