Sketch the Graph of g(x) = cos(x + π/4)
Question:
Sketch the graph of the following trigonometric function :
\[ g(x)=\cos\left(x+\frac{\pi}{4}\right) \]
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 left by \[ \frac{\pi}{4} \] units.
Important properties:
- Amplitude \(=1\)
- Period \(=2\pi\)
- Phase shift \(=\dfrac{\pi}{4}\) to the left
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\cos\frac{\pi}{4} =\frac{\sqrt2}{2}\\[8pt] x=\frac{\pi}{4} &\Rightarrow y=\cos\frac{\pi}{2}=0\\[8pt] x=\frac{3\pi}{4} &\Rightarrow y=\cos\pi=-1\\[8pt] x=\frac{5\pi}{4} &\Rightarrow y=\cos\frac{3\pi}{2}=0\\[8pt] x=\frac{7\pi}{4} &\Rightarrow y=\cos2\pi=1 \end{aligned} \]
Thus the curve passes through the points
\[ \left(0,\frac{\sqrt2}{2}\right),\quad \left(\frac{\pi}{4},0\right),\quad \left(\frac{3\pi}{4},-1\right),\quad \left(\frac{5\pi}{4},0\right),\quad \left(\frac{7\pi}{4},1\right) \]
Plot these points and draw a smooth cosine curve through them.
Hence, the required graph is shown above.
Graph Features:
- Amplitude = \(1\)
- Period = \(2\pi\)
- Phase shift = \(\dfrac{\pi}{4}\) to the left
- Range = \(-1 \le y \le 1\)