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I NEED HELP
Find the equation of the graph given below. Notice that the sine function is used in the answer template, representing a sine function that is shifted and/or reflected.

Use the variable x in your equation rather than the multiplication × symbol.
y = __ sin (___) + (___)

I NEED HELP Find The Equation Of The Graph Given Below Notice That The Sine Function Is Used In The Answer Template Representing A Sine Function That Is Shifted class=

Sagot :

Given:

The graph of a sine function.

To find:

The sine function.

Solution:

The general form of a sine function is

[tex]y=Asin(Bx+C)+D[/tex]            ...(i)

Where, |A| is amplitude, [tex]\dfrac{2\pi}{B}[/tex] is period, [tex]-\dfrac{C}{B}[/tex] is phase shift and D is mid-line.

From the given graph it is clear that the minimum value of the function is -4 and the maximum value is 0.

[tex]|A|=\dfrac{Maximum-Minimum}{2}[/tex]

[tex]|A|=\dfrac{0-(-4)}{2}[/tex]

[tex]|A|=\dfrac{4}{2}[/tex]

[tex]|A|=2[/tex]

The graph is reflected across the mid-line, so A=-2.

And,

[tex]D=\dfrac{Maximum+Minimum}{2}[/tex]

[tex]D=\dfrac{0+(-4)}{2}[/tex]

[tex]D=\dfrac{-4}{2}[/tex]

[tex]D=-2[/tex]

Period of the function is 4π because it completer its one cycle in the interval of 4π.

[tex]4\pi=\dfrac{2\pi}{B}[/tex]

[tex]B=\dfrac{2\pi}{4\pi}[/tex]

[tex]B=\dfrac{1}{2}[/tex]

Phase shift is [tex]\dfrac{\pi}{4}[/tex].

[tex]-\dfrac{C}{B}=\dfrac{\pi}{4}[/tex]

[tex]-\dfrac{C}{\frac{1}{2}}=\dfrac{\pi}{4}[/tex]

[tex]C=-\dfrac{1}{2}\times \dfrac{\pi}{4}[/tex]

[tex]C=-\dfrac{\pi}{8}[/tex]

Putting [tex]A=-2,B=\dfrac{1}{2},C=-\dfrac{\pi}{8},D=-2[/tex] in (i), we get

[tex]y=-2\sin\left(\dfrac{1}{2}x-\dfrac{\pi}{8}\right)-2[/tex]

Therefore, the required equation is [tex]y=-2\sin\left(\dfrac{1}{2}x-\dfrac{\pi}{8}\right)-2[/tex].

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