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.