ADV 122 GRAPHING RATIONAL FUNCTIONS ADV 122 GRAPHING

ADV 122 GRAPHING RATIONAL FUNCTIONS We have graphed several types functions, now we are

ADV 122 GRAPHING RATIONAL FUNCTIONS Pay attention to the transformation clues! (-a indicates a

ADV 122 GRAPHING RATIONAL FUNCTIONS Asymptotes n Places on the graph the function will

ADV 122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = x Vertical Asymptote: x =

ADV 122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = x+4 x + 4 indicates

ADV 122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = – 3 x+4 x +

ADV 122 GRAPHING RATIONAL FUNCTIONS Graph: f(x) = 1 +6 x+1 x + 1

ADV 122 GRAPHING RATIONAL FUNCTIONS How do we find asymptotes based on an equation

ADV 122 GRAPHING RATIONAL FUNCTIONS n Vertical Asymptotes (easy one)

ADV 122 GRAPHING RATIONAL FUNCTIONS Horizontal Asymptotes (H. A) n

ADV 122 GRAPHING RATIONAL FUNCTIONS 3 cases

ADV 122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator is GREATER than

ADV 122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator and numerator are

ADV 122 GRAPHING RATIONAL FUNCTIONS If there is a Vertical Shift n

ADV 122 GRAPHING RATIONAL FUNCTIONS Practice

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ADV 122 GRAPHING RATIONAL FUNCTIONS

ADV 122 GRAPHING RATIONAL FUNCTIONS We have graphed several types functions, now we are adding one more to the list! Graphing Rational Functions

ADV 122 GRAPHING RATIONAL FUNCTIONS

ADV 122 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.

ADV 122 GRAPHING RATIONAL FUNCTIONS Asymptotes n Places on the graph the function will approach, but will never touch.

ADV 122 GRAPHING RATIONAL FUNCTIONS 1 Graph: f(x) = x Vertical Asymptote: x = 0 Horizontal Asymptote: y = 0 No horizontal shift. No vertical shift. A HYPERBOLA!!

ADV 122 GRAPHING RATIONAL FUNCTIONS

ADV 122 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

ADV 122 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 =-3

ADV 122 GRAPHING RATIONAL FUNCTIONS Graph: f(x) = 1 +6 x+1 x + 1 indicates a shift 1 unit left Vertical Asymptote: x = -1 +6 indicates a shift 6 units up moving the horizontal asymptote to y = 6 Horizontal Asymptote: y = 6

ADV 122 GRAPHING RATIONAL FUNCTIONS

ADV 122 GRAPHING RATIONAL FUNCTIONS How do we find asymptotes based on an equation only?

ADV 122 GRAPHING RATIONAL FUNCTIONS n Vertical Asymptotes (easy one)

ADV 122 GRAPHING RATIONAL FUNCTIONS Horizontal Asymptotes (H. A) n

ADV 122 GRAPHING RATIONAL FUNCTIONS 3 cases

ADV 122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator is GREATER than the numerator. n The Asymptote is y=0 ( the x-axis)

ADV 122 GRAPHING RATIONAL FUNCTIONS If the degree of the denominator and numerator are the same: n

ADV 122 GRAPHING RATIONAL FUNCTIONS If there is a Vertical Shift n

ADV 122 GRAPHING RATIONAL FUNCTIONS Practice