# Equations and Graphs

A really useful skill to develop when learning mathematics is graph sketching; although some of the GCSE and A Level textbooks do have a short chapter covering this somewhere amongst the pages, unfortunately the value of graph sketching is, for the most part, almost invariably overlooked. Students think that this is something that they only need to do when they are asked to do it and even then it’s clear in many cases that they don’t really understand what it is that they are doing.

An equation such as $y=x^2-5x+7$ gives a clear, unambiguous, relationship between $x$ and $y$. So what? Well this means that for each $x$ value that I put into the right hand side of the equation I will get a corresponding y value. So for example if I choose $x=3$ (there is nothing special about this choice of value for x; I could just as easily have chosen $x=5.87$ or $x=-29$), I simply replace any $x$’s on the right hand side with $3$ to get $y=3^2-5×3+7$ and so $y=9-15+7$ and therefore $y=1$.

For many GCSE and A Level students this is fairly straightforward – but they miss the important part. This $x$ value and its corresponding $y$ value form a pair of coordinates, in this case $(3,1)$ – remember that the $x$ coordinate is the first number, $3$, and the $y$ coordinate is the second number, $1$.

Again – SO WHAT?

Well if I do this for several x values then I get a few more corresponding y values and so more pairs of coordinates which I can start to plot on some $xy$-axes. The more $xy$ pairs that I find the more points I can plot; what emerges is the graph of this equation. In many cases you don’t even need to plot a great deal of points to figure out what the shape of the graph is going to look like; many equations can be grouped together into families of equations – equations that have certain similarities – and the graphs of these families of equations (quadratic, exponential, logarithmic) have characteristic shapes and behaviours.

This is important – the equation and the graph represent the same thing but in different ways; the equation determines what the graph will look like and the graph will (at least in theory) determine the equation. If I choose any point on the graph and study the coordinates of that point I will find that the relationship between the $x$ and $y$ coordinates, for all points, is the relationship described by the equation and, moreover, any $x$ and $y$ values that are related via the equation WILL, without exception, be the coordinates of a point on the graph of that equation. So now, rather than seeing an equation as a string of miscellaneous symbols on a page I can start to draw (in some cases of my own volition) the graph of an equation and from this I can see (in the literal sense) the behaviour of the equation at a glance. The equation suddenly becomes more than an equation – it has more character to it. Drawing the graph of an equation might seem like more work – why would you inflict that on yourself? But that isn’t the case. By knowing what the graph of an equation looks like you get a visualisation of the situation and this usually means that overall you have don’t have to work anywhere near as hard to get your solutions.

Sadly, many GCSE and A Level maths students never get to grips with this. Not because they CAN’T draw graphs but because it is never made clear to them that this relationship between equation and graph exists. They go on thinking that in order to draw the graph of an equation they have to have been told beforehand what the graph of that particular equation looks like. This is a disaster for them! I often ask my students to sketch the graph of an equation and to start with they may look at me blankly or say, “I’ve never been told what that looks like”. My response might usually be something like, “Well think about what the equation is telling you is going to happen for different values of $x$”. This might not hit home straight away but eventually it does in most cases and, finally, they understand that the graph (even of unfamiliar equations) can be figured out themselves by thinking about what the equation is doing at various (well chosen) values of $x$.

Once you get your head around this then you’ll find that maths becomes ten times easier.