General Form

You have already seen how to graph a quadratic function when it is in its standard form. Recall that the standard form of a quadratic function looks like:

But how do you graph a quadratic function when it is in its general form? The general form of a quadratic function looks like this:

If you want to graph a quadratic function that is in general form, you have to convert it to standard form.

How do we convert to standard form? Take a look at the first example:

Example 1 - Converting the General Form of a Quadratic Equation to Standard Form

Convert the quadratic function to standard form.

Solution

First look to see if you can factor the trinomial easily. In this case, you cannot. So...on to the next step which is completing the square. Completing the square is a series of algebraic steps designed to simplify a quadratic function. You should have been shown this process in class, but you can use this page as a reference. The steps are listed below beside each algebraic step.

So y = (x + 2)² - 5 is the standard form of y = x² + 4x - 1. Now you can graph it using your knowledge of transformations.

Example 2

Convert the quadratic function to standard form.

Solution

This example is a little bit harder, as having a value other than 1 for a adds a bit of trouble to completing the square. Again, the steps are shown below, with a brief description beside each step.

Let's stop here to explore what just happened a little further. You add (b/2)² inside the brackets on the right hand side to make a perfect square trinomial. But you did not just add 1 to the right side. The 3 outside the brackets means that everything inside the brackets is multiplied by 3. Therefore, you just added three to the right hand side. That is why you need to add three to the left hand side (and not 1).

Now you have seen a couple of examples of converting general form to standard form, but how about the other way around?

Example 3 - Converting a Quadratic Function in Standard Form to General Form

We'll look at the function from example 2. Convert to general form.

Solution

To convert to general form, just expand the (x + 1)² and simplify the right hand side. The work is shown below:

You should make sure that you have all the notes for this topic. Your teacher will have covered this material in more detail. There are questions in your booklet, but you may also try a couple of sample questions, or you can head back to the Quadratic Functions main page.