Quadratics can be dealt with in two ways - algebraically and graphically.
A lot of students see these as two separate topics, however, they do cross over and you can use one technique when dealing with the other.
The significant points of a quadratic graph
All quadratic graphs have the same form, they look something like this:
For all quadratic graphs, there are three things you need to be able to find using an algebraic method - the roots, the y-intercept and the vertex.
In this worksheet, we’re going to look at how we find the vertex (the turning point) algebraically.
How to find the coordinates of the vertex (turning point) for the quadratic y = x2 + 3x - 4
For this, we have to use one of the techniques for dealing with quadratics, completing the square.
Step 1: Complete the square for the quadratic
Using the technique of completing the square, we can see that:
y = x2 + 3x – 4 → y = (x + 1.5)2 – 6.25
Step 2: Find the x coordinate of the turning point
This relies on you knowing something about square numbers – they can never be negative.
The smallest a squared number can be is 0.
When we are looking for the vertex, we are looking for where the value of y is as low as possible (the minimum) - this happens when the squared bracket is zero.
So what value of x will make the squared bracket zero?
(x + 1.5)2 = 0
x + 1.5 = 0
x = -1.5
Step 3: Find the y coordinate of the turning point
We’ve already worked out that the minimum point is at x = -1.5.
To find the value of y, all we have to do is to plug x = -1.5 into the completed square y = (x + 1.5)2 – 6.25
If we do this, we get y = (-1.5 + 1.5)2 – 6.25
The bracket now becomes zero and leaves us with y = -6.25
This means that the coordinates of the vertex are (-1.5,-6.25)
Is the vertex always the lowest point?
No. If you have a graph of the form y = x2 + ax + b you will get a minimum, however if you have a graph of the form y = -x2 + ax + b, you will get a maximum.
Let's try some questions now.