### Why do this problem?

This
problem provides a context to explore graphs of trigonometric
functions and look at the effects on the graph when the equation is
changed. Rather than asking learners to sketch the graphs from the
equations, they have to figure out the equation from the graph. The
problem gives the opportunity to investigate reflections, stretches
and translations of curves, and the corresponding effects on
equations.

### Possible approach

On plain paper or pupil whiteboards, sketch the purple curve
$y=\sin x$. Ask learners to mark on the coordinates of any
interesting points, such as intersections with the axes and turning
points.

Next, sketch the red curve on the same diagram and discuss in
pairs how to make the red curve from the purple one. How would the
graphs continue if the $x$ axis was continued past
$360^{\circ}$?

Now sketch the green curve and again mark on the key points.
Ask pairs to discuss similarities and differences between the
purple and green curves, and to consider how to transform one graph
into the other.

Reveal the information that the green curve has equation
$y=\sin 2x$, and set the challenge of working out the equations of
the other curves. Graphical calculators or graphing software could
be used to experiment with changing the equation and looking at the
effect on the graph.

### Key questions

What are the interesting features of the graph of $y=\sin
x$?

What happens to the graph beyond $360^{\circ}$?

How would you describe the effect on the graph when $x$ is
changed to $2x$? $3x$?

### Possible extension

Learners could create their own patterns using a combination
of sin, cos and tan graphs, and challenge others to work out the
equations.

Sine
problem gives another example of a pattern made from sine
graphs which learners could try to recreate.

### Possible support

Start by investigating transformations of more familiar graphs
in

Parabolic
Patterns and related problems.