How do you find a possible formula of the following form?

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Answer 1 · 2018-07-22T05:29:45

$y = sin t + 2$

Both $y = A sint + k$ and $y = A cos t + k$ have the same midline as $y = 0$ . The period is $2pi$ , for both. Amplitude is $abs A$ , for both.

Changing $y$ to $y- 2$ , the midline becomes $y - 2 = 0$ . So, $k = 2$ , for both.

The minimum amplitude $= 0$ , in the midline $y = 2$ . At $t = 0$ , $y = A sin t + 2 = 2$ . OK.

It is not so for . $y = A cos t + 2 = 3$ , with amplitude, the maximum $A$ . So change $t$ to $(t +- pi/2)$ . Then

$y = A cos(t +- pi/2) + 2 = 2$ , at $t = 0 rArr$ the amplitude is $0$ .

See graphs, with $A = 1$ .

Graph of $y = sin t + 2$ :

graph{(y-sin x -2)(y-2) = 0}

Graph of $y = cos ( t - pi/2 ) + 2$ :

graph{(y - cos ( x - pi/2 )-2)(y-2) = 0}

Graph of $y = cos ( t + pi/2 ) + 2$ :

graph{(y - cos ( x + pi/2 )-2)(y-2) = 0}

Observe that the first two are the same and the third graph is different.