Sketch the Graph of ψ(x) = 4 sin 3(x − π/4) for 0 ≤ x ≤ 2π
Question:
Sketch the graph of the following function :
\[ \psi(x)=4\sin 3\left(x-\frac{\pi}{4}\right),\quad 0 \le x \le 2\pi \]
Solution:
We know that the graph of
\[ y=\sin x \]
is a standard sine curve.
In the function
\[ y=4\sin 3\left(x-\frac{\pi}{4}\right) \]
- Amplitude \(=4\)
- Period \(=\dfrac{2\pi}{3}\)
- Phase shift \(=\dfrac{\pi}{4}\) units to the right
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=4\sin\left(-\frac{3\pi}{4}\right) =-2\sqrt2\\[8pt] x=\frac{\pi}{4} &\Rightarrow y=4\sin0=0\\[8pt] x=\frac{5\pi}{12} &\Rightarrow y=4\sin\frac{\pi}{2}=4\\[8pt] x=\frac{7\pi}{12} &\Rightarrow y=4\sin\pi=0\\[8pt] x=\frac{3\pi}{4} &\Rightarrow y=4\sin\frac{3\pi}{2}=-4 \end{aligned} \]
The graph repeats after every interval of
\[ \frac{2\pi}{3} \]
Thus similar waves continue up to \(x=2\pi\).
Plot the important points and draw smooth sine curves through them.
Hence, the required graph is shown above.
Graph Features:
- Amplitude = \(4\)
- Period = \(\dfrac{2\pi}{3}\)
- Phase shift = \(\dfrac{\pi}{4}\) to the right
- Domain = \(0 \le x \le 2\pi\)
- Range = \(-4 \le y \le 4\)