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Step-by-step explanation:
The equation of a trigonometric function is
or
Let define some variables,
D is the midline, this refers to the midpoint of the highest y value and lowest y value. Some textbooks call it the vertical translations but it is the same thing.
A is the amplitude. The amplitude is the distance from the midline to the highest y value. Some distance is non negative, the formula for the amplitude is
The period is how often the wave repeats itself on a interval.
Period can't be negative so the formula for period is
To find the period, look at the extreme points.
The phase shift tells us if the sinusoid have been shifted to the right or left. The formula for the phase shift
Solving for x gives us
If our x is negative, we have a phase shift to the right
If our x is Positve, we have a phase shift to the left.
Let solve this equation, Let use Sine since sin(0)=0,
The smallest y value here is 0, and the highest is 2, so the amplitude is 1.
Next, the distance from the max to the midline is 1, as well the min to the midline is also 1.
So
We have minimum at 0, and 8, so our period is 8.
Solve for b,
Plug this in the equation.
The graph passes through (4,2) so let see if that holds true for our equation
This doesn't hold true, so we must have a phase shift
Notice that
So if we shift this pi/2 to the right, we can get our equation to be true.
This equation works so our equation is