Sketch the Graphs of y = sin x and y = sin(x + π/4) on the Same Axes
Question:
Sketch the graphs of the following pairs of functions on the same axes:
\[ f(x)=\sin x \]
\[ g(x)=\sin\left(x+\frac{\pi}{4}\right) \]
Solution:
We know that
\[ y=\sin x \]
is the standard sine curve.
Now consider
\[ y=\sin\left(x+\frac{\pi}{4}\right) \]
This graph is obtained by shifting the graph of \[ y=\sin x \] to the left by \[ \frac{\pi}{4} \] units.
Both functions have:
- Amplitude \(=1\)
- Period \(=2\pi\)
Now calculate some important points.
For \(y=\sin x\)
\[ \begin{aligned} x=0 &\Rightarrow y=0\\[6pt] x=\frac{\pi}{2} &\Rightarrow y=1\\[6pt] x=\pi &\Rightarrow y=0\\[6pt] x=\frac{3\pi}{2} &\Rightarrow y=-1\\[6pt] x=2\pi &\Rightarrow y=0 \end{aligned} \]
For \(y=\sin\left(x+\frac{\pi}{4}\right)\)
\[ \begin{aligned} x=0 &\Rightarrow y=\sin\frac{\pi}{4}=\frac{\sqrt2}{2}\\[8pt] x=\frac{\pi}{4} &\Rightarrow y=\sin\frac{\pi}{2}=1\\[8pt] x=\frac{3\pi}{4} &\Rightarrow y=\sin\pi=0\\[8pt] x=\frac{5\pi}{4} &\Rightarrow y=\sin\frac{3\pi}{2}=-1\\[8pt] x=\frac{7\pi}{4} &\Rightarrow y=\sin2\pi=0 \end{aligned} \]
Plot these points and draw smooth sine curves through them.
The graph of \[ y=\sin\left(x+\frac{\pi}{4}\right) \] appears shifted to the left compared to \[ y=\sin x \]
Hence, the required graphs are shown above.
Graph Features:
- Both graphs have amplitude \(1\)
- Both graphs have period \(2\pi\)
- \(y=\sin(x+\pi/4)\) is shifted \(\pi/4\) units to the left