babeeb0oL

2021-08-22

Write an equation for a sinusoidal graph with the following properties:
$A=-3$ period $=\frac{2\pi }{3}$ phase shift $=-\frac{\pi }{4}$

rogreenhoxa8

Step 1
The general equation of sinusoidal equation is $y=A\mathrm{sin}\left(kx-\varphi \right)$ where A is the amplitude, C is the phase shift.
Equation the given period that is $\frac{2\pi }{3}$ to $\frac{2\pi }{k}$ to obtain the value of k.
$\frac{2\pi }{3}=\frac{2\pi }{k}$
$k=3$
Step 2
Equate the given phase shift to $\frac{\varphi }{k}$ and Substitute 3 for k in $-\frac{\pi }{4}=\frac{\varphi }{k}$ to obtain the value of $\varphi$
$-\frac{\pi }{4}=\frac{\varphi }{3}$
$\varphi =-\frac{3\pi }{4}$
Substitute the obtained values of in $y=A\mathrm{sin}\left(kx-\varphi \right)$ and simplify to obtain the equation of sinusoidal equation.
$y=A\mathrm{sin}\left(kx-\varphi \right)$
$y=-3\mathrm{sin}\left(3x-\left(-\frac{3\pi }{4}\right)\right)$
$=-3\mathrm{sin}\left(3x+\frac{3\pi }{4}\right)$

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