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