Sketch the Graphs of y = sin(x/2) and y = sin x on the Same Axes
Question:
Sketch the graphs of the following pairs of functions on the same axes:
\[ f(x)=\sin\frac{x}{2} \]
\[ g(x)=\sin x \]
Solution:
We know that
\[ y=\sin x \]
is the standard sine curve having period
\[ 2\pi \]
Now consider
\[ y=\sin\frac{x}{2} \]
Its period is
\[ \frac{2\pi}{1/2}=4\pi \]
Hence, the graph of \[ y=\sin\frac{x}{2} \] stretches horizontally and completes one wave in the interval \[ 0 \le x \le 4\pi \]
Both graphs have amplitude \(1\).
Important points for \[ y=\sin\frac{x}{2} \] are:
\[ (0,0),\quad (\pi,1),\quad (2\pi,0),\quad (3\pi,-1),\quad (4\pi,0) \]
Important points for \[ y=\sin x \] are:
\[ (0,0),\quad \left(\frac{\pi}{2},1\right),\quad (\pi,0),\quad \left(\frac{3\pi}{2},-1\right),\quad (2\pi,0) \]
and the pattern repeats up to \(4\pi\).
Plot these points and draw smooth sine curves on the same coordinate axes.
Hence, the required graphs are shown above.
Graph Features:
- Amplitude of both graphs = \(1\)
- Period of \(y=\sin x\) is \(2\pi\)
- Period of \(y=\sin(x/2)\) is \(4\pi\)
- \(y=\sin(x/2)\) is stretched horizontally compared to \(y=\sin x\)