Write an equation in terms of x and y for

Tahmid Knox 2021-08-18 Answered

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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Expert Answer

Jozlyn
Answered 2021-08-19 Author has 13136 answers

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

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