Answer:
y = 4sin(2x) +1
Step-by-step explanation:
When writing the equation for a sine or cosine function, you need to look at four things:
- amplitude
- period
- vertical offset
- horizontal offset
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Amplitude
The amplitude of a sine or cosine function is half the difference of the peak values. Here, the maximum is 5 and the minimum is -3, so the amplitude is ...
amplitude = 1/2(5 -(-3)) = 4
Period
The period is the horizontal distance between corresponding parts of the function. The peaks occur at π/4 and 5π/4, so the period is ...
period = (5π/4) -(π/4) = π
Vertical offset
The vertical offset is the midpoint between the highest and lowest values.
vertical offset = 1/2(5 +(-3)) = 1
Horizontal offset
The value here will depend on whether you want to write the function using sine or cosine. Here, we notice that the y-intercept is the same as the vertical offset, so this can be written as a sine function with no horizontal offset.
That is, the horizontal offset will be the horizontal location (x-value) of the peak of a cosine function, or of the midpoint of a sine function.
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Putting these together to write the equation, you use ...
y = (amplitude)��sine(2π/(period)×(x -horzontal offset)) +(vertical offset)
Using the values we found above:
- amplitude = 4
- period = π
- vertical offset = 1
- horizontal offset = 0
we have ...
y = 4sin(2x) +1
