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 $$\displaystyle{y}={a} \sin{{b}}{\left({x}-{d}\right)}+{c}$$ or $$\displaystyle{y}={a} \cos{{b}}{\left({x}-{d}\right)}+{c}$$, has: amplitude ∣a∣, period $$\displaystyle\frac{{{2}\pi}}{{b}}$$, vertical tration c units up if $$c>0$$ or $$∣c∣$$ 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:

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

The period is $$\displaystyle\pi$$ so P$$\displaystyle\frac{{{2}\pi}}{{b}}=\pi\rightarrow{2}$$. The amplitude is 3 so $$a=∣3∣=3$$

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

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

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