Sketching graphs: the DATA method

The DATA method is probably the coolest acronym I’ve ever come up with - it’s even about graphs! It’s a four-point plan for how to sketch a graph and be pretty sure of getting the salient features right.

D is for domain

The first thing to decide is, where is the function you’re trying to draw defined? Are you given a domain in the question? If it has square roots or logarithms in, figure out which values of $x$ you’re allowed to use. If you’re dividing by something that’s not constant, you need to make sure the bottom of the fraction isn’t zero, too - watch out for sneaky things like $\sec (x)$, which is really $\frac{1}{\cos (x)}$ and isn’t defined at $x=\frac{1}{2}(2n+1)\pi$.

A is for asymptotes

By this stage, you’ve done half the work on asymptotes - when you looked for denominators becoming zero, you found most of the vertical asymptotes. (You’d also get one if you had a ${\log }_a(x)$, and let $x$ go to zero). You need to think a bit about the behaviour on either side of asymptotes - just work out the sign of every factor and decide whether it ends up being positive or negative.

You’ve also got horizontal asymptotes: you get these by thinking about what happens when $x$ gets big (in both positive and negative directions). Does it go to infinity? Does it get very small? Does it tend to a particular value? If you’re feeling very advanced, does it approach any particular line or curve?

T is for turning points

You can find turning points, of course, by differentiating and finding what values of $x$ make the gradient zero. Use the second derivative to check whether it’s a maximum (negative second derivative is a frowny face), a minimum (positive second derivative is a smiley face) or a point of inflection (if the second derivative is also zero). Make sure you find the $y$-values, too!

A is for axes

Lastly - although it’s the first thing you learn to do in C1 sketching - where does the graph cross the $y$-axis (easy! what happens when $x=0$?) and the $x$-axis (a bit harder - you have to solve $y=0$, which can be trickier).

Once you have those elements in place, it’s just a case of joining things up! Remember, your sketch isn’t going to go in the National Gallery, so it doesn’t have to be a work of art: it just needs to show the important points and have the right sort of shape.

One last thing: draw BIG. The more space you have for labels, the clearer you can make things.

* Edited 2014-06-27 to correct where $\cos (x)=0$.