Sketch the Graphs of y = sin 2x and y = 2 sin x on the Same Axes
Question:
Sketch the graphs of the following pairs of functions on the same axes:
\[ f(x)=\sin 2x \]
\[ g(x)=2\sin x \]
Solution:
Consider the graph of
\[ y=\sin 2x \]
Its amplitude is \[ 1 \] and its period is \[ \frac{2\pi}{2}=\pi \]
Now consider the graph of
\[ y=2\sin x \]
Its amplitude is \[ 2 \] and its period is \[ 2\pi \]
Hence:
- \(y=\sin 2x\) oscillates faster
- \(y=2\sin x\) has greater height
Important points for \[ y=\sin 2x \] are:
\[ (0,0),\quad \left(\frac{\pi}{4},1\right),\quad \left(\frac{\pi}{2},0\right),\quad \left(\frac{3\pi}{4},-1\right),\quad (\pi,0) \]
and the pattern repeats up to \(2\pi\).
Important points for \[ y=2\sin x \] are:
\[ (0,0),\quad \left(\frac{\pi}{2},2\right),\quad (\pi,0),\quad \left(\frac{3\pi}{2},-2\right),\quad (2\pi,0) \]
Plot these points and draw smooth curves on the same coordinate axes.
Hence, the required graphs are shown above.
Graph Features:
- Amplitude of \(y=\sin 2x\) is \(1\)
- Amplitude of \(y=2\sin x\) is \(2\)
- Period of \(y=\sin 2x\) is \(\pi\)
- Period of \(y=2\sin x\) is \(2\pi\)
- \(y=\sin 2x\) oscillates faster while \(y=2\sin x\) has greater amplitude