Sketch the Graph of g(x) = 3 sin(x − π/4) for 0 ≤ x ≤ 5π/4
Question:
Sketch the graph of the following function :
\[ g(x)=3\sin\left(x-\frac{\pi}{4}\right), \quad 0 \le x \le \frac{5\pi}{4} \]
Solution:
We know that the graph of
\[ y=\sin x \]
is a standard sine curve.
In the function
\[ y=3\sin\left(x-\frac{\pi}{4}\right) \]
- Amplitude \(=3\)
- Phase shift \(=\frac{\pi}{4}\) units to the right
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=3\sin\left(-\frac{\pi}{4}\right) =-\frac{3\sqrt2}{2}\\[6pt] x=\frac{\pi}{4} &\Rightarrow y=3\sin0=0\\[6pt] x=\frac{3\pi}{4} &\Rightarrow y=3\sin\frac{\pi}{2}=3\\[6pt] x=\frac{5\pi}{4} &\Rightarrow y=3\sin\pi=0 \end{aligned} \]
Thus the curve passes through the points
\[ \left(0,-\frac{3\sqrt2}{2}\right),\quad \left(\frac{\pi}{4},0\right),\quad \left(\frac{3\pi}{4},3\right),\quad \left(\frac{5\pi}{4},0\right) \]
Plot these points and draw a smooth sine curve through them.
Hence, the required graph is shown above.
Graph Features:
- Amplitude = \(3\)
- Phase shift = \(\frac{\pi}{4}\) to the right
- Domain = \(0 \le x \le \frac{5\pi}{4}\)
- Range = \(-3 \le y \le 3\)