Transformations of Graphs
Like many things in life, graphs come in groups or categories. The process of
graphing a function can be simplified by knowing the basic structure of a particular graph
type and how it relates to the specific function. For example, all quadratic equations have
the same bowed shape. The only difference between the graphs of two different quadratic
equations is the eccentricity (degree) of the bow and the placement or location of the graph
on the coordinate system.
Because all graphs within a particular category are fundamentally the same, we can
change one graph into another. This process is called a transformation.
The transformation of a graph is a relocation of the points of the graph. There are
two ways that we can move a point. We can actually relocate the point or we can change
the axis system that we are measuring the point against (an in depth discussion of
transformations is in the appendix). For purposes of visual simplicity we shall talk and act
as if we are always moving the points of the graph. Under a given transformation the point
(x, y) will be transformed into a new point (x', y').
As a general rule, we can break transformations into two types.
Those which change the x coordinates of the graph and those which
change the y coordinates. The application of a transformation to the x
variable will affect the graph in a horizontal direction only. The
application of a transformation to the y variable will affect the graph in the
vertical direction only.
When dealing with function notation y = f(x), transformations on
the inside of the function symbol f() are applied to the x variable and
transformations applied outside the function symbol are applied to the y
variable.
Consider the function f(x) = x shown in graph 1. We want to see
what happens to the graph when we add or multiply values to the inside
of the function (directly on the x variable) or outside the function (applied
to the function rule itself). In order to see how the graph changes we will
look at the points (0, 0), (1, 1) and (–1, –1) on the graph of f(x).
Now look at the graph of the function y = f(x) + 2 (Graph 2).
Because f(x) = x this is equivalent to y = x + 2. The graph of this equation
is shown in graph 2. This transformation of f(x) added 2 units to the outside of the function
which is a change in the y direction. Looking at the graph we see that the point (0, 0) has
changed to the point (0, 2) or that 2 units have been added to the y coordinate. Similarly,
the point (1, 1) has become (1, 3) and the point (–1, –1) has become (–1, 1). Visually, the
graph is raised 2 units.
What happens when we add 2 units inside the function (to the x
variable) to get y = f(x + 2)? Again this is equivalent to the equation
y = x + 2. Only this time we are changing the graph in the x direction.
Now the point (0, 0) becomes (–2, 0). This is the same as subtracting 2
units from the x coordinate. Similarly, subtracting 2 from the x
coordinates of (1, 1) and (–1, –1) gives us (–3, –1) and (–1, 1)
respectively. Visually, the graph shifts 2 units in the negative direction
(see graph 3).
Translations
1
The student should show that subtracting 2 units outside the function lowers the
graph 2 units or subtracts 2 units from each of the y coordinates. Also, subtracting 2 units
inside the function shifts the graph 2 units in the positive x direction or adds 2 units to each
x coordinate.
Example 1: Consider the graph of f(x). Graph the function y = f(x – 3) + 1.
Under the transformation the basic shape of the graph will
remain the same. All we have to do is move a few points around to
determine the new placement.
The transformation adds a 1 to the outside of the function.
This means that we add 1 unit to each of the y coordinates of the
points on the graph. The transformation also subtracts a 3 on the
inside of the function. Here we add 3 units to each of the x
coordinates of the graph.
The point (–3, –3) becomes (0, –2),
(–1, 2) becomes (2, 3),
(1, 2) becomes (4, 3),
(2, 0) becomes (5, 1),
and (3, 3) becomes (6, 4).
Plot these points and connect them similar to the original graph to get the graph of
the transformed equation.
This type of transformation, which shifts the graph around the coordinate plane, is
called a translation. Another type of transformation, called a scaling, distorts the graph by
stretching or shrinking it with respect to the coordinate axes.
Refer again to the function f(x) = x. The graph of y = 2f(x) is shown in graph 4.
Because of the function definition, this is equivalent to y = 2x which is a line with slope 2
and y intercept at the origin.
To see how the graph has been changed look at the points (1, 1)
and (–1, –1) on the original graph. On the new graph the corresponding
points are (1, 2) and (–1, –2). To convert from the first pair of points to the
second we multiply the y coordinate by 2. Consequently, multiplying
outside the function by 2 multiplies the y coordinates by 2. This is the
same as stretching the graph by a factor of two in the y direction.
Translations
2
The graph of the equation y = f(2x) also reduces to y = 2x
because of the definition of f (see graph 5). Because the
transformation is inside the function symbol the transformation is
only in the x direction. This time the point (1,1) becomes the
point ( 12 , 1) and the point (–1, –1) becomes (– 12 , –1).
Multiplying on the inside of the function by 2 has reduced the x
coordinates by a factor of 2 (or multiplied the x coordinates by 12 ).
This is equivalent to shrinking the graph by 12 in the x direction.
Adding a units on the outside of the function raises the graph of the function a units
(adds a units to each y coordinate of the function).
Adding a units on the inside of the function moves the graph of the function –a
units (subtracts a units form each of the x coordinates).
Multiplying outside the function by a units stretches the graph by a factor of a in the
y direction (multiplies each y coordinate by a).
Multiplying inside the function by a units shrinks the graph by a factor of a
1
(multiplies each x coordinate by a ).
Notice that the transformation of the x coordinates is the reverse of what we might
expect while the transformation of the y coordinates are direct and straightforward.
Combining translations and scalings is a relatively simple procedure as long as we
follow a few simple rules.
If we wish to transform the function f(x) by y = af(cx + d) + b where a, b, c, and d are
constants, then for each point (x,y) on the original graph, the corresponding point (x', y') on
x–d
the transformed graph is found by the transformation equations y' = ay + b and x' =
c .
This latter equation is derived by letting x = cx' + d and then solving for x'.
These two rules follow our above discussion of transformations. To transform the y
coordinate we rescale by multiplying by a and translate by adding b. To transform the x
coordinate we translate by subtracting d and then rescale by dividing by c. Note carefully
the order of the operations. To transform the y coordinate we first rescale then translate.
To transform the x coordinate we first translate and then we rescale. This is in accord with
the 'backwards' process of transforming the x coordinate that we observed above.
Translations
3
Example 2: Consider the function y = p(x). Draw the graph of the transformed
function y = 13 p( 12 x – 1 ) + 2.
To transform the y coordinates of this graph we
let y' = 13 y + 2. To transform the x coordinates let x =
1
2
x' – 1, solve for x' to get x' = 2(x + 1) or x' = 2x + 2.
We then apply these two transformations to the points
(–2, 3), (–1, 0), (0, –3), (2, 0), and (4, –3) with the
following results:
(–2, 3)  (–2, 3)
(–1, 0)  (0, 2)
(0, –3)  (2, 1)
(2, 0)  (6, 2)
(4, –3)  (10, 1).
Problems
1.
Given the following graph of y = f(x), graph
a. y = f(x + 4)
b. y = f(x) + 4
c. y = 2f(x)
d. y = f(2x)
e. y = 2f(2x + 4) + 4
2.
Given the following graph of y = g(x), graph
a. y = –g(x)
b. y = g(–x)
c. y = –g(x) + 2
d. y = –g(x – 2)
e. y = g(–x – 2)
Translations
4
3.
Given the following graph of h(x), graph
a. y = 12 h(x)
b. y = h(12 x)
c. y = 12 h(12 x)
d. y = – 12 h(–2x)
e. y = 2h(–x + 2) – 3
4.
Given the following graph of S(x), graph
a. y = S(x) + 1
b. y = 2S(x)
c. y = S( 2x – )

d. y = 12 S(3x + 2 )
e. y = 2S(12 x + 2)
5.
Given the following graph of R(x), graph
a. y = R(x) + 1
b. y = R(x – 2)
c. y = R(x – 2) + 1
d. y = 2R(x)
e. y = R(3x)
f. y = 2R(3x – 3) + 2
Notice that R(x) is not a function. It fails the vertical line test for functions. However,
our transformations still work.
Translations
5
Answers
1.
(–8, 0)
(–6, –2)
(–2, 4)
(2, 2)
(4, –6)
a
(–12, 0)
(–10, –2)
(–6, 4)
(–2, 2)
(0, –6)
b
(–8, 4)
(–6, 2)
(–2, 8)
(2, 6)
(4, –2)
c
(–8, 0)
(–6, –4)
(–2, 8)
(2, 4)
(4, –12)
d
(–4, 0)
(–3, –2)
(–1, 4)
(1, 2)
(2, –6)
e
(–6, 4)
(–5, 0)
(–3, 12)
(–1, 8)
(0, –8)
2.
a
(–8, 0)
(–5, –3)
(–4, 0)
(–3, 3)
(3, –3)
b
(8, 0)
(5, 3)
(4, 0)
(3, –3)
(–3, 3)
c
(–8, 2)
(–5, –1)
(–4, 2)
(–3, 5)
(3, –1)
(–7, 0)
a
(–7, 0)
b
(–14, 0)
c
(–14, 0)
(–5, 2)
(–5, 1)
(–10, 2)
(–10, 1)
(–3, 0)
(–3, 0)
(–6, 0)
(–6, 0)
(0, 3)
(0,
(0, 3)
(0,
(5, –2)
(5, –1)
(10, –2)
(10, –1)
(7, 0)
(7, 0)
(14, 0)
(14, 0)
(–, 0)
a
(–, 1)
b
(–, 0)
c
(0, 0)
(–8, 0)
(–5, 3)
(–4, 0)
(–3, –3)
(3, 3)
d
(–6, 0)
(–3, –3)
(–2, 0)
(–1, 3)
(5, –3)
e
(6, 0)
(3, 3)
(2, 0)
(1, –3)
(–5, 3)
d
e
(9, –3)
3.
3
2
)
3
2
)
( 72 , 0)
( 52 , –1)
( 32 , 0)
(0, – 32 )
(– 52 , 1)
(– 10
2 , 0)
(7, 1)
(5, –3)
(2, 3)
(–3, –7)
(–5, –3)
4.
(–

2
, –1)
(–

2
, 0)
(–

2
, –2)
( 4 , –1)
(0, 0)
(0, 1)
(0, 0)
( 2 , 0)
( 2 , 1)
( 2 , 2)
( 2 , 2)
( 3
4 , 1)
(, 0)
(, 1)
(, 0)
(, 0)
Translations
6
d
(–
(–

3
(–

2
(0,
(–
, 0)
,–

6

6
e
(–6, 0)
1
2
)
(–5, –2)
, 0)
(–4, 0)
1
2
)
(–3, 2)
, 0)
(–2, 0)
5.
(–2, 0)
a
(–2, 1)
b
(0, 0)
c
(0, 1)
d
(–2, 0)
( 13
(0, 2)
(2, 0)
(0, 3)
(2, 1)
(2, 2)
(4, 0)
(2, 3)
(4, 1)
(0, 4)
(2, 0)
(1, 6)
( 53 , 2)
(0, –2)
(0, –1)
(2, –2)
(2, –1)
(0, –4)
(1, –2)
Translations
7
e
, 2)