Quixotic Quadratic Queries Qause Qonfusion

Welcome to Mathematical Methods – a subject which will prepare you for using mathematics to analyse the world and introduce the concept of developing theoretical models to investigate potential results of decisions. It is also a subject that will require you to work hard to master the skills and techniques so that you can use them effectively.

We have already completed the simplest work of the year, revising skills which you should be familiar with from previous years of mathematics – simple linear equations, systems of functions and basic coordinate geometry. Now, we start to go further, by investigating non-linear relationships – that means all lines that can be drawn on a graph that are not straight. The simplest of these are the quadratics – which have the shape of a smile (or a frown), just like the one in graph just above.

You will have studied this last year, and you may have become quite good at factorising and sketching quadratics – which is good. If you are not so confident, you will have time to revise, but you will have to put in additional work. We will be going a lot further with quadratics over the next month, and you will need to be prepared!

So, first let’s define a quadratic. You may remember we defined a polynomial in class recently, as an expression where each term contains a positive integer power of a pronumeral. You can see the sort of expressions which are not polynomials in the picture to the right. A quadratic is a particular type of pronumeral, where the highest power is “2” – in other words, it has an “x²” term in the expression, and nothing higher.

You’ll need to be able to do many algebraic operations on quadratic equations, but the first thing is converting between the three different forms of any quadratic equation. The first form is the so called “standard form”, which has the form of “y = ax² + bx + c”. This form is the most useless – the only information you can get is the y-intercept (from the “c” constant) and the general shape (from the “a” constant). You can see more about the effects of these constants at this link.

The second form is the factorised form, in which you see the expression rewritten and re-organised in brackets, i.e. “y = a(x + b)(x + c). Note that the values a a,b & c are not the same in any of the forms, except by coincidence. From the factorised form, you can use the null-factor law to determine the x-intercepts of the graph of the equation. Note that it may be possible that a particular quadratic function does not have a factorised form (if the graph does not have x-intercepts!).

The third and final form is called either the “vertex form” or “turning – point form), and has the following general appearance: y = a(x + b)² + c, and as you might guess by it’s name, it gives you the coordinates of the turning point of the graph. You can get to the vertex form by completing the square on the standard form.

Here is a video that covers the same theme, but with a voice-over explanation:

Finally, here are three documents that may help you:

1. Factoring Flowchart (link)

2. Quadratic Notes (link)

3. Advanced Notes (link)

And also checklist for chapter two (link).

Now, it’s your turn. Share a resource that you have found – on the internet, or anywhere else – that you have found useful so far this year – and an explanation of why you have found it useful. *Everyone* in the class must share a resource. If you find a resource that someone else posts useful – say so!

See you in class!

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