Graph Transform notes
Keywords: Translations , Reflections, Stretches
Consider the graph of y=x2, shown below.
x
y
-2
4
-1
2
0
0
1
1
2
2
The word translation means (in maths at least!) to move in a straight line. If we
wish to translate this curve (equation) in the +/- y direction then we add or
subtract the required movement. So for example y= x2+2 and y= x2-2 will
translate the curve up and down 2 units in the Y axes.
f(x)=x2+2
f(x)=x2-2
+
Figure 1: y=x2
Figure 2:y=x2+2 and y=x^2-2
Note : In your exam it is likely that the notation used will refer to y=f(x), where
f(x)= x2, or more simply f(x)= x2. This notation is convenient as it allows the
graph transformations in a convenient manner as we will see.
Summary 1- We can think of the graph for y=x2+a, as y-a= x2 and our y-axes
moving down by a units.
This type of translation is
f(x)→f(x)+a
a translation of the graph by a units in the y-axes
Translations in the x-axes
Below is a graph of the equation y=x3, or as we are now going to refer to it as
f(x)=x3.
If we want to move this 3 units in x direction then we need to plot the
equation f(x)=(x-3)3 and to move -3 units in the x axes we plot f(x)=(x+3)3.
(X+3)3
Figure 3: y=f(x); f(x)=x3
(X-3)3
Figure 4: y=f(x); f(x)=(x-3)3; f(x)=(x+3)3
Summary 2The transformation
f(x)→f(x-a)
The transformation
f(x)→f(x+a)
Is a translation of a units along the x
axes
Is a translation of -a units along the x
axes
Below is a question lifted from an A-level paper, it looks much worse than it is
because of the algebra involved. But this should be good practice for you!
Straight away you know
that the shape of the
graphs will be the same
Because f(x)→f(x+a) is
a translation of -a
units in the x axes.
This is why the f(x)
notation is so useful
Stretches
Consider the graph of y=f(x) ; f(x)=x2+2x-1, as shown in the diagram below. We
are going to see what the transformations f(2x) and 2f(x) do to the shape of
the graph.
The equations are:
f(x)
x2+2x-1
f(0.5x)
0.25x2+x-1
2f(x)
2x2+4x-2
The graphs are shown below, firstly y=2f(x)
The dotted line is y=2f(x).
The solid line is y=f(x)
If you look carefully the graph of y=f(x)
has been stretched by a factor of 2 in
the y direction. I have included two
sets of points to help illustrate this.
The graph of y=αf(x) is a stretch
parallel to the y axes by a factor of α
The graph of y=f(0.5x) has a different effect. This can be seen in the diagram
overleaf.
The graphs are shown below, firstly y=f(0.5x)
The dotted line is y=f(0.5x).
The solid line is y=f(x)
If you look carefully the
graph of y=f(x) has been
stretched by a factor of 2 in
the x direction. I have
included two sets of points
to help illustrate this.
The graph of y=f(αx) is a
stretch of factor 1/α
parallel to the x axes
Reflections
Consider the graph y=f(x), f(x)=x3+1, it is shown in the picture below.
We will now look at the transformations, y=-f(x) and y=f(-x)
y=f(x)
x3+1
y=-f(x)
- x3-1
Y=f(-x)
- x3-1
The graphs are shown overleaf
y=-f(x)
The dotted line is the graph of y=-f(x)
The solid line is the graph of y=f(x)
Imagine the x-axes to be a mirror. The
graph of y=-f(x) can then be seen to be
a reflection in the x-axes
The function y=-f(x) is a reflection of
y=f(x) in the x axes
y=f(-x)
The dotted line is the graph of y=f(-x)
The solid line is the graph of y=f(x)
Imagine the y-axes to be a mirror. The
graph of y=f(-x) can then be seen to be
a reflection in the y-axes.
The function y=f(-x) is a reflection of
y=f(x) in the y axes