Transformation of
Functions
Homework 7
Vertical Shift
Let 𝑓 be a function and c a positive real
number.
The graph of 𝑦 = 𝑓 𝑥 + 𝑐 is the graph of
𝑦 = 𝑓(𝑥) shifted c units vertically upward.
The graph of 𝑦 = 𝑓 𝑥 − 𝑐 is the graph of
𝑦 = 𝑓(𝑥) shifted c units vertically
downward.
Vertical shift upward: two units
Figure 1.47
Sketch the graph
𝑔 𝑥 = 𝑥 3 +1
Horizontal Shift
Let f be a function and c a positive real
number.
The graph of 𝑦 = 𝑓(𝑥 + 𝑐) is the graph of
𝑦 = 𝑓(𝑥) shifted to the left c units.
The graph of 𝑦 = 𝑓(𝑥 − 𝑐) is the graph of
𝑦 = 𝑓(𝑥) shifted to the right c units.
Horizontal shift to the right: two units
Figure 1.48
Sketch the graph
𝑔 𝑥 = (𝑥 + 1)3
Reflections
Reflection about the x-Axis
The graph of y = – f(x) is the graph of y = f(x) reflected about the x-axis.
Reflection about the y-Axis
The graph of y = f(–x) is the graph of y = f(x) reflected about the y-axis.
Examples of reflections
Write the Equation
The shape of 𝑓 𝑥 = 𝑥 2 , but shifted two units to the left, nine units
up, and then reflected in the x-axis.
Write the Equation
The shape of 𝑓 𝑥 = 𝑥 , but shifted four units to the right, six units
down, and then reflected in the y-axis.
Rigid vs Nonrigid Transformations
Horizontal shifts, vertical shifts, and reflections are rigid transformations
because the basic shape of the graph is unchanged.
These transformations change only the position of the graph in the
coordinate plane.
Nonrigid transformations are those that cause a distortion—a change in
the shape of the original graph.
Vertical Stretches and Shrinks
A nonrigid transformation of the graph of
y = f (x) is represented by g(x) = cf (x),
where the transformation is
• a vertical stretch when c > 1
• a vertical shrink when 0 < c < 1
Vertical Stretches and Shrinks
Horizontal Shrinks and Stretches
Another nonrigid transformation of the
graph of y = f (x) is represented by h(x) = f
(cx), where the transformation is
• a horizontal shrink when c > 1
• a horizontal stretch when 0 < c < 1
Horizontal Shrinks and Stretches
|3𝑥|
1
𝑥
3
List the parent functions and
transformations
𝑔 𝑥 =− 𝑥+1 3
List the parent functions and
transformations
𝑔 𝑥 =3 𝑥−4 +2
List the parent functions and
transformations
𝑔 𝑥 = −3 − 𝑥 + 5 2
List the parent functions and
transformations
𝑔 𝑥 =
1
𝑥−3+4
2
Order of Transformations
When transforming one function into another, apply each
transformation in this order:
1. Horizontal shifting
2. Stretching or shrinking
3. Reflecting
4. Vertical shifting
Graph a Transformation
Use the graph of f(x) = x2 to obtain the graph of 𝑓 𝑥 = − 𝑥 − 1 2 + 3