12. Answer: y = 2 sin(4x) + 1
Step-by-step explanation:
The general form of a sin equation is: y = A sin (Bx - C) + D
[tex]A=\dfrac{max - min}{2}\\\\\\.\ =\dfrac{3-(-1)}{2}\\\\\\.\ =\dfrac{4}{2}\\\\\\.\ =2[/tex]
D = max - A
= 3 - 2
= 1
[tex]\text{The period of the given graph is } \dfrac{\pi}{2}\ and\ \text{the formula for period is }\dfrac{2\pi}{B}\\\\\rightarrow \dfrac{2\pi}{B}=\dfrac{\pi}{2}\\\\\\\text{Cross multiply: }4\pi=B\pi\\\\\text{Divide both sides by }\pi: B=4[/tex]
There is no phase shift so C = 0
A = 2, B = 4, C = 0, D = 1 --> Equation of the graph is: y = 2 sin (4x) + 1
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12. Answer: [tex]\bold{y=\dfrac{1}{2}\ cos\bigg(\dfrac{1}{2}x\bigg)}[/tex]
Step-by-step explanation:
The general form of a cos equation is: y = A cos (Bx - C) + D
[tex]A=\dfrac{max - min}{2}\\\\\\.\ =\dfrac{\frac{1}{2}-(-\frac{1}{2})}{2}\\\\\\.\ =\dfrac{1}{2}[/tex]
[tex]\text{D = max - A}\\\\.\ =\dfrac{1}{2}-\dfrac{1}{2}\\\\.\ =0[/tex]
[tex]\text{The period of the given graph is}\ 4\pi \ and\ \text{the formula for period is }\dfrac{2\pi}{B}\\\\\rightarrow \dfrac{2\pi}{B}=4\pi\\\\\\\text{Cross multiply: }2\pi=4\pi B\\\\\text{Divide both sides by }4\pi: B=\dfrac{1}{2}[/tex]
There is no phase shift so C = 0
A = [tex]\dfrac{1}{2}[/tex], B = [tex]\dfrac{1}{2}[/tex], C = 0, D = 0 --> Equation of the graph is: [tex]y=\dfrac{1}{2}\ cos\bigg(\dfrac{1}{2}x\bigg)[/tex]
