Challenge Level

**Why do this problem?**

The problem gives a context for investigating the periodic behaviour of functions involving sines, cosines and the modulus function and discovering the effects of combining these functions. Points where the first derivative is not defined occur and are clearly illustrated by the graph. It is instructive for learners to realise that although they can find the derivative on both sides of a point,
if it takes different values on each side then the derivative is undefined and there is no tangent at that point.

**Possible approach**

Learners can use graph plotters to plot the graphs (such as Desmos or Geogebra) and then explain the form and features of the graph, making the task easier but perhaps not so rewarding. If they want more of a challenge they can analyse the equations, sketch the graphs and then use a graph
plotter to check their findings.

Many useful issues for class discussion arise from this problem, such as how to write down the equation of a function which takes different values on different intervals, how to interpret the behaviour of the function where the derivative is undefined, the amplitude of oscillations etc.

**Key questions**

When does the sine function take positive values and when is it negative?

When the derivative of a function at one side of a point has a different value to the derivative on the other side what happens to the tangent to a graph at that point?

What is the significance of $A$ and $\alpha$ in the graph of the function $f(x) = A\sin (x + \alpha)$?