# Write an equation in terms of x and y for

Write an equation in terms of x and y for the function that is described by the given characteristics. A sine curve with a period of $\pi$, an amplitude of 3, a right phase shift of $\frac{\pi }{2}$, and a vertical translation up 2 units.

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Jozlyn

The graph of $y=a\mathrm{sin}b\left(x-d\right)+c$ or $y=a\mathrm{cos}b\left(x-d\right)+c$, has: amplitude ∣a∣, period $\frac{2\pi }{b}$, vertical tration c units up if $c>0$ or $\mid c\mid$ units down if $c<0$, and phase shift dd units to the right if $d>0$ or ∣d∣ units to the left if $d<0$.

Use the sine function:

$y=a\mathrm{sin}b\left(x-d\right)+c$

The period is $\pi$ so P$\frac{2\pi }{b}=\pi \to 2$. The amplitude is 3 so $a=\mid 3\mid =3$

The phase shift is $\frac{\pi }{2}$ right so $d=\frac{\pi }{2}$. The vertical traation is 2 units so d=2. So, the equation is:

$y=3\mathrm{sin}2\left(x-\frac{\pi }{2}\right)+2$

or $y=3\mathrm{sin}\left(2x-\pi \right)+2$