Download - Transformations of the Parent Functions
![Page 1: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/1.jpg)
Transformations of the
Parent Functions
![Page 2: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/2.jpg)
What is a Parent Function
A parent function is the most basic version of an algebraic function.
![Page 3: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/3.jpg)
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|
![Page 4: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/4.jpg)
Types of TransformationsVertical Translations
Vertical S t r e t c h
Vertical Compression
Reflections
Over the x-axis
![Page 5: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/5.jpg)
….More TransformationsHorizontal TranslationsHorizontal S t r e t c hHorizontal CompressionReflections
Over the y-axis
![Page 6: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/6.jpg)
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.
![Page 7: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/7.jpg)
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.
![Page 8: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/8.jpg)
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
![Page 9: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/9.jpg)
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
![Page 10: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/10.jpg)
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.
![Page 11: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/11.jpg)
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
![Page 12: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/12.jpg)
Multiple TransformationsGraph a function involving more than one transformation in the following order:
Horizontal translation Stretching or compressing Reflecting Vertical translation
![Page 13: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/13.jpg)
Are we there yet?Parent FunctionsFunction Families
TransformationsMultiple Transformations
InversesAsymptotes
![Page 14: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/14.jpg)
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.
![Page 15: Transformations of the Parent Functions](https://reader036.vdocuments.net/reader036/viewer/2022062305/568164db550346895dd732e9/html5/thumbnails/15.jpg)
AsymptotesBoundary line that a graph will not cross.Vertical AsymptotesHorizontal AsymptotesAsymptotes adjust with the transformations of the parent functions.