Sketch the Graph of θ(x) = sin(x/2 − π/4) for 0 ≤ x ≤ 4π
Question:
Sketch the graph of the following function :
\[ \theta(x)=\sin\left(\frac{x}{2}-\frac{\pi}{4}\right), \quad 0 \le x \le 4\pi \]
Solution:
We know that the graph of
\[ y=\sin x \]
is a standard sine curve.
In the function
\[ y=\sin\left(\frac{x}{2}-\frac{\pi}{4}\right) \]
- Amplitude \(=1\)
- Period \(=\dfrac{2\pi}{1/2}=4\pi\)
- Phase shift \(=\dfrac{\pi/4}{1/2}=\dfrac{\pi}{2}\) units to the right
Thus one complete sine wave is obtained in the interval
\[ 0 \le x \le 4\pi \]
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\sin\left(-\frac{\pi}{4}\right) =-\frac{\sqrt2}{2}\\[8pt] x=\frac{\pi}{2} &\Rightarrow y=\sin0=0\\[8pt] x=\frac{3\pi}{2} &\Rightarrow y=\sin\frac{\pi}{2}=1\\[8pt] x=\frac{5\pi}{2} &\Rightarrow y=\sin\pi=0\\[8pt] x=\frac{7\pi}{2} &\Rightarrow y=\sin\frac{3\pi}{2}=-1\\[8pt] x=4\pi &\Rightarrow y=\sin\left(\frac{7\pi}{4}\right) =-\frac{\sqrt2}{2} \end{aligned} \]
Thus the curve passes through the points
\[ \left(0,-\frac{\sqrt2}{2}\right),\quad \left(\frac{\pi}{2},0\right),\quad \left(\frac{3\pi}{2},1\right),\quad \left(\frac{5\pi}{2},0\right),\quad \left(\frac{7\pi}{2},-1\right) \]
Plot these points and draw a smooth sine curve through them.
Hence, the required graph is shown above.
Graph Features:
- Amplitude = \(1\)
- Period = \(4\pi\)
- Phase shift = \(\dfrac{\pi}{2}\) to the right
- Domain = \(0 \le x \le 4\pi\)
- Range = \(-1 \le y \le 1\)