Sketch the graph of f(x) = sin(x). On the same set of axes, draw the graph of f(x)+2, f(2x) and f(-x). By observing your graphs of f(x) and f(x), if f(a)=1, what is the value of f(-a)?

The first stage is the simple drawing of the sin graph. (could be done on the whiteboard)Then it is a typical A-Level style question to perform three types of transformations on that graph, here we have 1. the example of a translation 2 units upwards, 2. a compression in the x axis direction of scale factor 1/2, that is, sin(x) usually first crosses the positive x-axis at pi, now it crosses the positive x-axis at pi/2. 3. Then we have a reflection of the graph in the y axis. I would get these all drawn, can draw them on the whiteboard, can then label specific notable points, such as roots. For the final part, it is a matter of noticing that sin(-x)=-sin(x), and realising from here that f(-a) would be -1. Points of extending the question: From here we could practice using the chain rule, differentiating sin(2x), and lead onto a discussion about the double angle formulas, and differentiating various functions using the chain rule. Also, could try translating different graphs as well, including quadratics and cubics, and observing the new position of each of the roots, and turning points (again bringing in differentiation.)