In this section we relate graphs using transformations , which are functions that act on the xcoordinates and y-coordinates of points in order to change graphs in predictable ways. We put transformations into two categories, rigid and non-rigid. A transformation that distorts the shape of a graph is called non-rigid (for example, a vertical stretch), while a transformation that leaves the size and shape of a graph the same is called rigid (for example, reflections and translations).
A hint that I give students for remembering which direction a transformation is occurring in is to look at whether the transformation is occurring inside or outside of the function.
If a transformation is happening inside the function it is part of the input and will occur in the horizontal (or x) direction. Note: these transformations will do the opposite of what we intuitively think they will do!
If a transformation is happening outside the function it is part of the output and will occur in the vertical (or y) direction.
Transformation Rules:
Let c be a ____________ real number.
______________ a translation to the right by c units
______________ a translation to the left by c units
______________ a translation up by c units
______________ a translation down by c units
______________ across the x -axis
______________ across the y -axis
______________ a a
by a factor of
by a factor of
1 𝑐
1
if c > 1 if c < 1 𝑐
______________ a a
by a factor of
by a factor of c c
if if c c > 1.
< 1

Example 1: Translations
Describe how the graph of 𝑦 = |𝑥| can be transformed to the graph of the given equation.
(a) 𝑦 = |𝑥| − 4
(b) 𝑦 = |𝑥 + 2|
Example 2: Finding Equations for Translations
The figures below show the graph 𝑦 = 𝑥 3
(thin curve) and the graph of its transformation
(thicker curve). Write an equation for each transformation. (Note: A standard viewing window was used.)
Example 3 – Finding Equations for Reflections
Find the equation for the reflection of
5𝑥−9 𝑥 2 +3
across each axis.
(b) Reflection across the x-axis: (a) Reflection across the y-axis:

Example 4 – Graphing Absolute Value Compositions 𝑓(|𝑥|) – replace the portion of the graph to the left of the y -axis by a reflection of the portion to the right of the y -axis across the y -axis.
|𝑓(𝑥)| – reflect the portion of the graph below the x -axis across the x -axis, leaving the portion above it unchanged.
Refer to the graph of 𝑦 = 𝑓(𝑥) shown below. In each case, determine the graph of the new function.
(a) 𝑦 = |𝑓(𝑥)|
(b) 𝑦 = 𝑓(|𝑥|)
(c) 𝑦 = −|𝑓(𝑥)|
(d) 𝑦 = |𝑓(|𝑥|)|
Graph 1 _____ Graph 2 _____ Graph 3 _____ Graph 4 _____
Example 5 – Finding Equations for Stretches and Shrinks
Let 𝑦
1
= x
3
– 16x. Find equations for the following non-rigid transformations of 𝑦
1
.
(a) A vertical stretch by a factor of 3. (b) A horizontal shrink by a factor of ½.

Example 6 – Combining Transformations in Order
The graph of y = x
2
undergoes the following transformations, in order. Find the equation of the graph that results.
 a horizontal shift 2 units to the right
 a vertical stretch by a factor of 3
 a vertical translation 5 units up
If you apply the transformations in the opposite order, will you end up with the same graph?
Prove your answer algebraically.