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.

**Example: **

**How to find the coordinates of the vertex (turning point) for the quadratic y = x ^{2} + 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 = x^{2} + 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 = x^{2} + ax + b you will get a minimum, however if you have a graph of the form y = -x^{2} + ax + b, you will get a maximum.

Let's try some questions now.