Sketch the Graphs of y = cos 2x and y = cos(x − π/4) on the Same Axes
Question:
Sketch the graphs of the following curves on the same scale and the same axes:
\[ y=\cos2x \]
\[ y=\cos\left(x-\frac{\pi}{4}\right) \]
Solution:
Consider the graph of
\[ y=\cos2x \]
Its amplitude is \(1\) and its period is
\[ \frac{2\pi}{2}=\pi \]
Hence it completes two waves in the interval \[ 0 \le x \le 2\pi \]
Now consider the graph of
\[ y=\cos\left(x-\frac{\pi}{4}\right) \]
This is the standard cosine curve shifted to the right by
\[ \frac{\pi}{4} \]
Its amplitude is \(1\) and period is \(2\pi\).
Important points for \[ y=\cos2x \] are:
\[ (0,1),\quad \left(\frac{\pi}{4},0\right),\quad \left(\frac{\pi}{2},-1\right),\quad \left(\frac{3\pi}{4},0\right),\quad (\pi,1) \]
and the pattern repeats up to \(2\pi\).
Important points for \[ y=\cos\left(x-\frac{\pi}{4}\right) \] are:
\[ \left(0,\frac{\sqrt2}{2}\right),\quad \left(\frac{\pi}{4},1\right),\quad \left(\frac{3\pi}{4},0\right),\quad \left(\frac{5\pi}{4},-1\right),\quad \left(\frac{7\pi}{4},0\right) \]
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 \(y=\cos2x\) is \(\pi\)
- Period of \(y=\cos(x-\pi/4)\) is \(2\pi\)
- \(y=\cos(x-\pi/4)\) is shifted \(\pi/4\) units to the right