Graphing Standard Function & Transformations
A few standard graphs
f(x) = x2
f(x) = x
f(x) = x1/2
f(x) = 1
x
f(x) =│x│
f(x) = x3
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Graphing Standard Function & Transformations
The rules below take these standard plots and shift them horizontally/
vertically
Vertical Shifts
Let f be the 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
upwards.
The graph of y = f(x) - c is the graph of y = f(x) shifted c units vertically
downwards.
g(x) = x2 + 2 = f(x) + 2
h(x) = x2 – 3 = f(x) – 3
Look for the positive and negative sign. Positive sign makes the graph
move upwards and the negative sign makes it move downwards
Here is a picture of the graph of g(x) = x2 1. It is obtained from the graph of f(x) =
x2 by shifting it down 1 unit.
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UASP Student Success Centers
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Graphing Standard Function & Transformations
Horizontal Shifts
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.
g(x) = (x-3)2 = f (x-3)
h (x) = (x + 2)2 = f (x+2)
Here is a picture of the graph of g(x) = |x4|. It is obtained from the graph of f(x) =
|x| by shifting it to the right 4 units.
Horizontal/ Vertical Scaling
Horizontal Scaling
Let g(x) = f(cx) where c is a positive real number.
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Graphing Standard Function & Transformations
• If c > 1, the graph of g is the
graph of f, compressed in the x-
direction by a factor of c.
• If 0 < c < 1, then the graph is
stretched in the x-direction by a factor
of 1/c
Here is a picture of the graph of
g(x) = (0.5x)3. Since c = 0.5 < 1, the
graph is obtained from that of
f(x) = x3 by stretching it in the x-
direction by a factor of 1/c = 2.
Vertical Scaling
Let g(x) = cf(x) here c is a positive real number.
• If c > 1, the graph of g is the graph of f, stretched in the y-direction by a
factor of c.
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Graphing Standard Function & Transformations
• If 0 < c < 1, then the graph is compressed in the y-direction by a factor of
1/c.
Here is a picture of the graph of g(x) = 3(x)1/2. Since c = 3 > 1, the graph is
obtained from that of f(x) = x1/2 by stretching it in the y-direction by a factor of c
= 3.
Reflection about the x axis
The graph of y = - f (x) is the graph of y = f (x) reflected about the x- axis.
Here is a picture of the graph of g(x) = (x 2 1). It is obtained from the graph of f(x)
= x 2 1 by reflecting it in the x-axis.
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Graphing Standard Function & Transformations
Reflection about the y axis
The graph of y = f (-x) is the graph of y = f (x) reflected about the y-axis.
Here is a picture of the graph of g(x) =(0.5x)3+1. It is obtained from the graph of
f(x) = 0.5x3+1 by reflecting it in the y-axis.
Summary of Transformations
To graph Draw the graph of f and: Changes in the equation of
y = f(x)
Vertical Shifts Raise the graph of f by c units C is added to f (x)
y = f (x) + c
y = f (x) – c Lower the graph of f by c units C is subtracted from f (x)
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UASP Student Success Centers
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Graphing Standard Function & Transformations
Horizontal Shifts
y = f (x + c) Shift the graph of f to the left c units x is replaced with x + c
y = f (x – c) Shift the graph of f to the right c units x is replaced with x – c
Reflection about the
x axis Reflects the graph of f about the f (x) is multiplied by –1
y = - f(x) x axis
Reflection about the X is replaced with –x
y axis Reflect the graph of f about the
y = f(-x) y axis
Sample Question:
Sketch the curve for g(x) =
Solve for yourself:
Created by
UASP Student Success Centers
success.asu.edu | 480-965-9072
Graphing Standard Function & Transformations
Created by
UASP Student Success Centers
success.asu.edu | 480-965-9072
