Keywords: Translations , Reflections, Stretches

Graph Transform notes

Keywords: Translations , Reflections, Stretches

Consider the graph of y=x2, shown below.

x / -2 / -1 / 0 / 1 / 2
y / 4 / 2 / 0 / 1 / 2

The word translation means (in maths at least!) to move in a straight line. If we wish totranslate 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.

+
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= x2and 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.

Figure 3: y=f(x); f(x)=x3 /
Figure 4: y=f(x); f(x)=(x-3)3; f(x)=(x+3)3
Summary 2-
The transformation
f(x)→f(x-a)
Is a translation of a units along the x axes / The transformation
f(x)→f(x+a)
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!

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