transformations of the parent functions
Post on 23-Feb-2016
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Transformations of the
Parent Functions
What is a Parent Function
A parent function is the most basic version of an algebraic function.
Types of Parent FunctionsLinear f(x) = mx + bQuadratic f(x) = x2
Square Root f(x) = √xExponentialf(x) = bx
Rational f(x) = 1/xLogarithmicf(x) = logbx
Absolute Value f(x) = |x|
Types of TransformationsVertical Translations
Vertical S t r e t c h
Vertical Compression
Reflections
Over the x-axis
….More TransformationsHorizontal TranslationsHorizontal S t r e t c hHorizontal CompressionReflections
Over the y-axis
FAMILIES TRAVEL TOGETHER……
Families of Functions If a, h, and k are real numbers with a=0, then the graph of y = a f(x–h)+k is a transformation of the graph of y = f ( x).
All of the transformations of a function form a family of functions.
F(x) = (a - h)+ k – Transformations should be applied from the “inside – out” order.
Horizontal TranslationsIf h > 0, then the graph of y = f (x – h) is a translation of h units to the RIGHT of the graph of the parent function.
Example: f(x) = ( x – 3)
If h<0,then the graph of y=f(x–h) is a translation of |h| units to the LEFT of the graph of parent function.Example: f(x) = (x + 4)
*Remember the actual transformation is (x-h), and subtracting a negative is the same as addition.
Vertical TranslationsIf k>0, then the graph of y=f(x)+k is a translation of k units UP of the graph of y = f (x).
Example: f(x) = x2 + 3
If k<0, then the graph of y=f(x)+k is a translation of |k| units DOWN of the graph of y = f ( x).
Example: f(x) = x2 - 4
Vertical Stretch or Compression
The graph of y = a f( x) is obtained from the graph of the parent function by: stretching the graph of y = f ( x) by a when a > 1. Example: f(x) = 3x2
compressing the graph of y=f(x) by a when 0<a<1. Example: f(x) = 1/2x2
ReflectionsThe graph of y = -a f(x) is reflected over the y-axis.The graph of y = f(-x) is reflected over the x-axis.
Transformations - Summarized
Y = a f( x-h) + kVertical S t r e t c h or compression
Horizontal Translation
Vertical Translation
Horizontal S t r e t c h
or compression
Multiple TransformationsGraph a function involving more than one transformation in the following order:
Horizontal translation Stretching or compressing Reflecting Vertical translation
Are we there yet?Parent FunctionsFunction Families
TransformationsMultiple Transformations
InversesAsymptotes
Where do we go from here?
Inverses of functionsInverse functions are reflected over the y = x line.When given a table of values, interchange the x and y values to find the coordinates of an inverse function.When given an equation, interchange the x and y variables, and solve for y.
AsymptotesBoundary line that a graph will not cross.Vertical AsymptotesHorizontal AsymptotesAsymptotes adjust with the transformations of the parent functions.
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