How would you sketch the curve of a graph?

A frequent question the comes up in the early parts of core maths is; how would you sketch the graph of a function f(x)?

Sketching just means that you would have to find some key points on a graph - which are actually quite easy to find.

1. What is the domain of the graph?

The domain is just a fancy word for what values you are ALLOWED to put into the function, f(x). To do this you can simply look at the function and for example, see if are there any ways that a nasty "division by 0" can happen, if so, then the value where this happens is not allowed in the domain! So be careful around this point when sketching your graph. (You might also want to think about what happens when your numbers get very close to this? Does your graph shoot off to infinity?).

2. Does the graph meet the x or y axis?

Well to find this out you would firstly set x to 0 (if you can), what is the value of y that comes out? This is the value where the curve meets the y axis! You could then set y to 0 and find where the curve meets the x axis (be careful there could be multiple points).

3. Is there any turning points?

A turning point is where the graph is flat, i.e the gradient is 0. So to find this point you would differentiate and then set the dy/dx term to 0. Solve this and you have where there is a turning point!

But is it a "maximum" or a "minimum" point?

To find this out you could take some points that are just either side of the turning point you just found and put these points back into f(x). This should tell you whether these points are above or below the turning point, if they are above it is a maxima, they're below you have a maxima. Or you might have one below and one above, in this case you have a cubic graph!

Now you have a whole bunch of little proporties on your graph its time to sketch. You know whether there are any points where the function doesn't exist, they points where the functuon meets the axis and any turning points (aslo whether these points are a maxima or minima).

Now you know what your graph roughly looks like!!