GRAPHS OF RATIONAL FUNCTIONS F-BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

RATIONAL EXPRESSION  The quotient of two polynomial expressions where the denominator cannot be equal to zero  Graphs of rational functions can be continuous (no jumps or breaks in the graph) or discontinuous

POINTS OF DISCONTINUITY A point in the domain of the function for which there is no corresponding point in the range Non-removable: occurs when the graph has vertical asymptotes Removable: occurs when the numerator and denominator have a factor in common (hole in the graph)

PARTS OF A RATIONAL FUNCTION Asymptote: an imaginary line that the graph approaches but never crosses  Horizontal, vertical, or oblique (slant) Intercepts: where graph crosses x-axis and y- axis

FACTORING REVIEW  Check for a GCF first  Other types of factoring:  Difference of Squares: Two terms, perfect squares, minus sign in middle  Backwards FOIL: Three terms with no leading coefficient  Bottoms Up: Three terms with a leading coefficient

FINDING PARTS OF A RATIONAL FUNCTION  Factor the numerator and denominator  Identify any holes in the graph (same factor in numerator and denominator)  Find x and y intercepts  X-int: Set numerator = 0  Y-int: Substitute 0 for x  Identify any asymptotes  Sketch the graph to determine the domain

IDENTIFYING ASYMPTOTES

IDENTIFY ANY HOLES, ASYMPTOTES, AND INTERCEPTS

GRAPH EACH OF THE FOLLOWING AND IDENTIFY THE DOMAIN AND RANGE

TRANSFORMATIONS OF RATIONAL GRAPHS

If a is negative: reflection over x-axis If a > 1: vertical stretch (graph moves “away” from asymptotes) If a < 1: vertical shrink (graph moves “closer” to asymptotes) If h is positive (x – h): vertical asymptote shifts to the left If h is negative (x + h): vertical asymptote shifts to the right If k is positive (+ k): horizontal asymptote shifts up If k is negative (- k): horizontal asymptote shifts down

DESCRIBE THE TRANSFORMATIONS AND THEN SKETCH THE GRAPH