### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

### Weekly Challenges

The NRICH Stage 5 weekly challenges are shorter problems aimed at Post-16 students or enthusiastic younger students. There are 52 of them.

# How to Sketch Graphs

##### Stage: 5

Published February 2014.

Over the years that I've worked in schools, I've noticed an aversion to graph sketching among many Key Stage 4 and 5 students. It's not just me; when I talk to teachers and share classroom resources with algebraic and graphical approaches, many teachers tell me "Oh, MY kids won't use graphs, they hate sketching graphs, if they can't solve it algebraically they won't bother!"

If this is you, you're missing out - graph sketching is a skill that is well worth the effort it takes to learn. A quick sketch of a graph can illuminate a situation in a way that would not be so clear from a few pages of algebra. They do say that a picture is worth a thousand words, after all. I used to dislike graph sketching, and avoided it wherever possible when I did my A levels, but found it invaluable while studying undergraduate maths. I wish I'd practised sooner...

Make it big!
One of the most common mistakes with sketch graphs is to draw tiny little things, as if you're scared of putting pencil to paper. Make your sketches big enough to show detail.

It's a sketch, not an accurate drawing!
Don't get too hung up on getting your sketch 100% accurate and to scale. As long as the salient features are there and clearly labelled, your sketch graph is fine.

Annotate.
Write down the information you have used in order to determine the important features of the sketch. For example, if you calculated turning points, asymptotes, or intercepts, explicitly write down next to your diagram the implications of your calculations.

Now on to some particulars.
Many people have an internal checklist of things to work out before starting to sketch graphs. It doesn't really matter what order you work them out in, but you should consider the following:
• What happens when $x=0$? When $y=0$?
• What happens when $x \to \pm \infty$?
• Are there any asymptotes? Horizontal? Vertical? Oblique?
• Where are the turning points?
• Is it periodic?
• Is it an odd function? An even function?
Often, answering the above questions can be enough to give a good feel for the shape of a graph. Once you have marked on all of the above information, you can often do a 'join the dots' exercise to give a reasonable sketch.

Take a look at the video below, where I sketch $y=\frac{2}{3}\cos ^2 x$ and $y=\sin x$, the first part of STEP I 2008 question 4. You may wish to have a go at sketching the two graphs for yourself, before watching to see how I did it.

A common mistake on this type of question are to confuse the graph of $y=\cos ^2 x$ and the graph $y=|\cos x|$, as both are similar to $y=\cos x$ but always positive. Unlike the modulus graph, which suddenly flips at the x axis giving a sharp point, $\cos ^2$ is smooth and has a turning point. It is differentiable at this point, unlike the modulus graph.

Why not have a go at sketching some graphs of your own? You may find it helpful to check your answers using graphing software - GeoGebra and Desmos are both free to use.

To see a worked example of a graph-sketching STEP question, see STEP I 2010 q2 on our Worked Examples page.