Question

Given f(t) = -2sin(1/2t - pi/2) - 1, find the amplitude, midline, period, horizontal shift and sketch a graph with labeled axis’s

Answer 1

Solution:

Given:

$f(t)=-2\sin (\frac{1}{2}t-\frac{\pi}{2})-1$

To get the wave parameters, we compare it to the wave equation.

$\begin{gathered} y=A\sin (wt\pm kx) \\ y=A\sin (2\pi ft-\frac{2\pi x}{\lambda}) \end{gathered}$

Comparing both equations, the following can be deduced;

$\begin{gathered} y=A\sin (2\pi ft-\frac{2\pi x}{\lambda}) \\ f(t)=-2\sin (\frac{1}{2}t-\frac{\pi}{2})-1 \\ \\ \text{Amplitude, A = 2} \\ \text{Midline is at y = -1} \end{gathered}$

To get the period,

$\begin{gathered} 2\pi ft=\frac{1}{2}t \\ f=\frac{t}{2(2\pi t)} \\ f=\frac{1}{4\pi} \\ \\ R\text{ecall the formula relating frequency and period,} \\ T=\frac{1}{f} \\ T=\frac{1}{\frac{1}{4\pi}} \\ T=4\pi \\ \\ \text{The period is 4}\pi \end{gathered}$

The horizontal shift is;

$\frac{\pi}{2}$

Hence,

$\begin{gathered} A=2\ldots\ldots\ldots\ldots.\ldots\ldots..\ldots\text{Amplitude} \\ y=-1\ldots\ldots\ldots.\ldots..\ldots\ldots.midl\text{ine} \\ T=4\pi\ldots\ldots\ldots\ldots\ldots\ldots........period \\ Shift=\frac{\pi}{2}\ldots.\ldots\ldots\ldots\ldots..horizontal\text{ shift} \end{gathered}$

The graph is as shown below;