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Warm Up Graph the function . We have graphed several functions, now we are adding one more to the list! Graphing Rational Functions. Parent Function: . Pay attention to the transformation clues!. (-a indicates a reflection in the x-axis). a x – h. f(x) = + k. - PowerPoint PPT Presentation
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GRAPHING RATIONAL FUNCTIONSADV122
Warm Up
Graph the function
GRAPHING RATIONAL FUNCTIONSADV122
We have graphed several functions, now we are adding one more to the list!
Graphing Rational Functions
GRAPHING RATIONAL FUNCTIONSADV122
Parent Function:
GRAPHING RATIONAL FUNCTIONSADV122
f(x) = + kax – h
(-a indicates a reflection in the x-axis)
vertical translation(-k = down, +k = up)
horizontal translation(+h = left, -h = right)
Pay attention to the transformation clues!
Watch the negative sign!! If h = -2 it will appear as x + 2.
GRAPHING RATIONAL FUNCTIONSADV122
Asymptotes
Places on the graph the function will approach, but will never touch.
GRAPHING RATIONAL FUNCTIONSADV122
f(x) =
1x
Vertical Asymptote: x = 0Horizontal Asymptote: y = 0
Graph:
A HYPERBOLA!!
No horizontal shift.No vertical shift.
GRAPHING RATIONAL FUNCTIONSADV122
W look like?
GRAPHING RATIONAL FUNCTIONSADV122
Graph: f(x) = 1x + 4
Vertical Asymptote: x = -4
x + 4 indicates a shift 4 units left
Horizontal Asymptote: y = 0
No vertical shift
GRAPHING RATIONAL FUNCTIONSADV122
Graph: f(x) = – 31x + 4
x + 4 indicates a shift 4 units left
–3 indicates a shift 3 units down which becomes the new horizontal asymptote y = -3.
Vertical Asymptote: x = -4
Horizontal Asymptote: y = 0
GRAPHING RATIONAL FUNCTIONSADV122
Graph: f(x) = + 6x
x – 1
x – 1 indicates a shift 1 unit right
+6 indicates a shift 6 units up moving the horizontal asymptote to y = 6
Vertical Asymptote: x = 1
Horizontal Asymptote: y = 1
GRAPHING RATIONAL FUNCTIONSADV122
You try!!
2.
GRAPHING RATIONAL FUNCTIONSADV122
How do we find asymptotes based on an equation only?
GRAPHING RATIONAL FUNCTIONSADV122
Vertical Asymptotes (easy one) Set the denominator equal to zero
and solve for x. Example:
x-3=0 x=3
So: 3 is a vertical asymptote.
GRAPHING RATIONAL FUNCTIONSADV122
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: No H.A because Has a H.A because 3=3. Has a H.A because
GRAPHING RATIONAL FUNCTIONSADV122
3 cases
GRAPHING RATIONAL FUNCTIONSADV122
If the degree of the denominator is GREATER
than the numerator.
The Asymptote is y=0 ( the x-axis)
GRAPHING RATIONAL FUNCTIONSADV122
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.
Example: Asymptote is 6/3 =2.
GRAPHING RATIONAL FUNCTIONSADV122
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)
Example:
Vertical shift is 7, so H.A is at 7.
GRAPHING RATIONAL FUNCTIONSADV122
Homework
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Graphing%20Simple%20Rational%20Functions.pdf