The most important question you need to ask here is not about quadratics, it’s what does the word **sketch **actually mean?

A sketch has to be neat, done in pencil but **doesn’t **have to be measured.

So, when you sketch a quadratic, you don’t have to find all the values in a table, put these on etc.

You would have to do that if the question asked you to **draw **or** plot **the quadratic.

**So can I draw any old graph then?**

In a word, no. When you sketch a quadratic, it has to look like a quadratic graph and have the significant points roughly in the correct places and labelled with their exact value.

**What are the significant points?**

There are **three **significant points that you have to be able to find algebraically - the **roots **(solutions), the **turning point **(the vertex/minimum) and the **y-intercept**.

**For the quadratic function y = f(x):**

**The roots** are found by solving the equation f(x) = 0

**The vertex** is found by completing the square.

**The intercept** is found by putting x = 0 into the function.

If you are unsure about the methods to either solve the quadratic or complete the square, look at the activities **7956 **and **7957** on using algebra to find roots and turning points before you attempt this activity.

**Example: **

**Sketch the graph for the function y = x ^{2} – 6x + 8**

**The roots:** We could either use the quadratic formula or factorisation here to solve this.

Factorisation is quicker, so here goes:

On the x-axis, y = 0, so y = x^{2} – 6x + 8 →** x ^{2} – 6x + 8 = 0**

x^{2} – 6x + 8 = 0 → **(x – 2)(x – 4) = 0**

Solving this gives **x = 2 and x = 4**

As coordinates these are **(2,0)**** **and **(4,0)**

**The vertex:** We start this process by **completing the square **for the function y = x^{2} – 6x + 8

y = x^{2} – 6x + 8 → **y = (x - 3) ^{2} – 1**

To find the minimum point, we need to find the lowest value of y. Because a square cannot be negative, the smallest value happens when **(x - 3) ^{2 }= 0**

This gives the x value for the minimum point as x = 3.

When **x = 3, y = -1.** The minimum point of the graph is **(3,-1)**

**The y-intercept**

The y-intercept is where the line crosses the y–axis. At this point x = 0.

Putting x = 0 into the equation y = x^{2} – 6x + 8 gives **y = 8.**

The intercept is therefore at** (0,8)**

**Putting this onto the graph**

**Step 1:** Plot the points roughly onto a set of axes - notice that at this point, the graph has no numbers.

**Step 2:** Draw a smooth curve through these points.

**Step 3:** Label the significant points.

Let's move on to some questions.