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ADV122 GRAPHING RATIONAL FUNCTIONS Warm Up Graph the function =− +3 2 +4 ADV122 GRAPHING RATIONAL FUNCTIONS We have graphed several functions, now we are adding one more to the list! Graphing Rational Functions ADV122 GRAPHING RATIONAL FUNCTIONS Parent Function: = ADV122 GRAPHING RATIONAL FUNCTIONS Pay attention to the transformation clues! (-a indicates a reflection in the x-axis) a f(x) = +k x–h vertical translation (-k = down, +k = up) horizontal translation (+h = left, -h = right) Watch the negative sign!! If h = -2 it will appear as x + 2. ADV122 GRAPHING RATIONAL FUNCTIONS Asymptotes Places on the graph the function will approach, but will never touch. ADV122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = x Vertical Asymptote: x = 0 Horizontal Asymptote: y = 0 No horizontal shift. No vertical shift. A HYPERBOLA!! ADV122 GRAPHING RATIONAL FUNCTIONS W = − look like? ADV122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = x+4 x + 4 indicates a shift 4 units left Vertical Asymptote: x = -4 No vertical shift Horizontal Asymptote: y = 0 ADV122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = –3 x+4 x + 4 indicates a shift 4 units left Vertical Asymptote: x = -4 –3 indicates a shift 3 units down which becomes the new horizontal asymptote y = -3. Horizontal Asymptote: y = 0 ADV122 GRAPHING RATIONAL FUNCTIONS Graph: f(x) = x +6 x–1 x – 1 indicates a shift 1 unit right Vertical Asymptote: x = 1 +6 indicates a shift 6 units up moving the horizontal asymptote to y = 6 Horizontal Asymptote: y = 1 ADV122 GRAPHING RATIONAL FUNCTIONS You try!! 1 1. = +2 2. = 1 +3 −4 ADV122 GRAPHING RATIONAL FUNCTIONS How do we find asymptotes based on an equation only? ADV122 GRAPHING RATIONAL FUNCTIONS Vertical Asymptotes (easy one) Set the denominator equal to zero and solve for x. Example: = 6 −3 x-3=0 x=3 So: 3 is a vertical asymptote. ADV122 GRAPHING RATIONAL FUNCTIONS Horizontal Asymptotes (H.A) In order to have a horizontal asymptote, the degree of the denominator must be the same, or greater than the degree in the numerator. Examples: 2 −3 = +7 3 −2 = 3 −2 +1 = 2 No H.A because 2 > 1 Has a H.A because 3=3. Has a H.A because 1 < 2 ADV122 GRAPHING RATIONAL FUNCTIONS 3 cases ADV122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator is GREATER than the numerator. The Asymptote is y=0 ( the x-axis) ADV122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator and numerator are the same: Divide the leading coefficient of the numerator by the leading coefficient of the denominator in order to find the horizontal asymptote. 6 3 3 3 −2 Example: = Asymptote is 6/3 =2. ADV122 GRAPHING RATIONAL FUNCTIONS If there is a Vertical Shift The asymptote will be the same number as the vertical shift. (think about why this is based on the examples we did with graphs) 5 +7 −3 Example: Vertical shift is 7, so H.A is at 7. ADV122 GRAPHING RATIONAL FUNCTIONS Homework http://www.kutasoftware.com/FreeWorksheets /Alg2Worksheets/Graphing%20Simple%20Rati onal%20Functions.pdf