Graph Transformations | Best GCSE Guide
Introduction
What we are going to look at in this article is the transformation of graphs. For GCSE maths, you need to be able to describe the transformation of a curve in both the x and y axes.
As well as this you also need to be able to understand what is meant by a combined transformation as well as being able to use vector notation to describe a graph transformations.
Graph Transformations - Introduction
Starting with a curve y=x^2 we will apply a number of graph transformations in the x and y axis and look at the effect that this has on the equation of the curve.
- y=f(x) \mp a
Here we have the curve y=x^2
This type of curve is a quadratic curve and it is symmetrical about the y=axis which is the same as the line x = 0. Now let us suppose that we move or transform this equation by moving up in the positive y direction.
So let us say we now want to draw the curve y=x^2+2
What will this look like?
You can see from the diagram that the original curve has gone up 2 units in the positive y direction. So in terms of a vector translation, nothing has happened in the x-directon but the curve has moved 2 units in the y direction. As a vector this can be written as \left(\frac{0}{2}\right)
So in general we can say that a curve of the form y=f(x) \mp a represents a transformation of \mp aunits in the y-direction.
Graph Transformation y=f(x \mp a)
Again let us start off with the curve y=x^2
Suppose now we want to move the curve along the x-axis only and let us say by 2 units in the positive x – direction.
What will our equation look like now?
Before we answer this, as a vector this would be written as \left(\frac{2}{0}\right)
o here, x = 2. Now we will set this equal to zero so x – 2 = 0.
And it is this x – 2 that we will write in replacement for x. So our new equation is now y=(x-2)^2
It is surprising that even though the curve is moving in the positive x – direction, 2 is actually being subtracted from x and the graph below shows the result.
As we can see the original curve has moved two units to the right of the original.
A good way to determine the direction in which a curve moves is to set the brackets equal to 0. So with y=(x+2)^2, setting x+ 2 = 0 gives x = – 2. And this gives you the direction of the curve.
So in general we can say that a curve of the form y=f(x \pm a) represents a transformation of \mp a units in the x-direction.
Graph Transformations – Combined transformation
Starting with the curve y=x^2 suppose we want to transform the curve by \left(\frac{-2}{3}\right).
What will happen to the curve and what will this look like?
In terms of the x – axis, x = -2, so x + 2 = 0. In other words the curve is moving 2 units to the left in the x-direction and it is moving 3 units in the y-direction.
We can then write the new equation as y=(x+2)^2+3 and the graph is shown below.
From the diagram above you can see that this combined transformation the new curve as a new minimum point at (-2, 3)
Transformations and generic graphs
Suppose we have a generic curve such as y = f(x) and this is shown below.
From the curve you can see that there is a point (1, 1) which appears at a maximum point on the curve. We want to determine the new coordinates under the following transformations:
\begin{aligned} & y=f(x)+2 \\ & y=f(x)-3 \\ & y=f(x-2) \\ & y=f(x+3) \\ & y=f(x-1)+4 \end{aligned}Answers:
Here the curve moves up 2 units in the y-direction so the new coordinates will be (1,3)
Here the curve moves down 3 units in the y-direction so the new coordinates will be (1, -2)
Here the curve moves 2 units in the x-direction so the new coordinates will be (3,1)
Here the curve moves -3 units the x-direction so the new coordinates will be (-2, 1)
Here the curve moves 1 unit in the x-direction and then 4 units in the y direction so the new coordinates will be (2,5)