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The family of graphs of x^n + y^n =1 (for even n) includes the circle. Why do the graphs look more and more square as n increases?

Tangled Trig Graphs

Age 16 to 18 Challenge Level:

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.