Question

Determine the amplitude and the period for the function. Sketch the graph of the function over one period. $$ y=\sin \left(x+\frac{\pi}{2}\right) $$

Short answer

The amplitude for the function \(y = \sin \left(x + \frac{\pi}{2}\right)\) is \(A = 1\), and the period is \(T = 2\pi\). The graph of the function can be sketched using key points at (0, 1), \(\left(\frac{\pi}{4}, \frac{\sqrt{2}}{2}\right)\), \(\left(\frac{\pi}{2}, 0\right)\), \(\left(\frac{3\pi}{4}, -\frac{\sqrt{2}}{2}\right)\), and \((\pi, -1)\).

Step-by-step solution

Find the amplitude

The amplitude of the function is the coefficient of the sine function, which in this given function is 1. Therefore, the amplitude is: \[A = 1\]

Find the period

The period of the sine function is given by \(2\pi\). As there is no coefficient change inside the sine function, the period remains the same: \[T = 2\pi\]

Find key points in the function graph

To sketch the graph over one period, we need to find the following points: 1. Start of the period 2. Peak point 3. Zero crossing at the middle of the period 4. Valley point 5. End of the period For the given function \(y = \sin \left(x + \frac{\pi}{2}\right)\), the points are: 1. Start of the period: \(x = 0\), \(y = \sin(\frac{\pi}{2}) = 1\) 2. Peak point: \(x = \frac{\pi}{4}\), \(y = \sin(\frac{3\pi}{4}) = \sqrt{2}/2\) 3. Zero crossing at the middle of the period: \(x=\frac{\pi}{2}\), \(y=\sin(\pi)=0\) 4. Valley point: \(x=\frac{3\pi}{4}\), \(y = \sin(\frac{5\pi}{4}) = -\sqrt{2}/2\) 5. End of the period: \(x = \pi\), \(y = \sin(\frac{3\pi}{2}) = -1\)

Sketch the graph over one period

Now, using the key points found in the previous step, we can plot the graph of the function over one period. The graph begins at the point (0, 1), reaches its peak at \((\frac{\pi}{4}, \frac{\sqrt{2}}{2})\), crosses zero at \((\frac{\pi}{2}, 0)\), reaches the valley at \((\frac{3\pi}{4}, -\frac{\sqrt{2}}{2})\), and ends the period at \((\pi, -1)\), before repeating itself.