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 + b

Quadratic f(x) = x2

Square Root f(x) = √x

Exponentialf(x) = bx

Rational f(x) = 1/x

Logarithmicf(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 Transformations

Horizontal Translations

Horizontal S t r e t c h

Horizontal Compression

Reflections

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 Translations

If 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 Translations

If 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

Reflections

The 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 Transformations

Graph a function involving more than one transformation in the following order:

Horizontal translation

Stretching or compressing

Reflecting

Vertical translation

Are we there yet?

Parent Functions

Function Families

Transformations

Multiple Transformations

Inverses

Asymptotes

Where do we go from here?

Inverses of functions

Inverse 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.

Asymptotes

Boundary line that a graph will not cross.

Vertical Asymptotes

Horizontal Asymptotes

Asymptotes adjust with the transformations of the parent functions.