Sketch the Graphs of y = cos x 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:
\[ y=\cos x \]
\[ y=\cos\left(x-\frac{\pi}{4}\right) \]
Solution:
We know that
\[ y=\cos x \]
is the standard cosine curve.
The graph of
\[ y=\cos\left(x-\frac{\pi}{4}\right) \]
is obtained by shifting the graph of \[ y=\cos x \] to the right by \[ \frac{\pi}{4} \] units.
Both graphs have:
- Amplitude \(=1\)
- Period \(=2\pi\)
Important points for \[ y=\cos x \] are:
\[ (0,1),\quad \left(\frac{\pi}{2},0\right),\quad (\pi,-1),\quad \left(\frac{3\pi}{2},0\right),\quad (2\pi,1) \]
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:
- Both graphs have amplitude \(1\)
- Both graphs have period \(2\pi\)
- \(y=\cos(x-\pi/4)\) is shifted \(\pi/4\) units to the right