state the equation of this function

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The maximum is 40 and the minimum is 20; the period is `90^@` or `pi/2` .

The cosine wave begins at its maximum -- this curve begins at its minimum so it is a transformation of `y=-cosx` .

The general form is `y=acosb(x-h)+k`

a: gives the amplitude (if a<0 reflects across horizontal line of symmetry.)

b: found by `b=(2pi)/p` where p is the period

h: phase shift or horizontal translation

k: vertical translation

The amplitude is 20. (`A=("max"-"min")/2` ). So a=-20.

b is 4 (`(2pi)/(pi/2)=4` )

If the base function is the cosine there is no phase shift so h=0.

k=20. ( y=k is the midline found by `k=("max"+"min")/2` )

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Therefore the function is `y=-20cos4x+20`

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If you think the base function is the sine, you must use a phase shift of `pi/8` or `22.5^@` to the left.

The function would be `y=-20sin(4(x+pi/8))+20` or `y=-20sin(4x+pi/2)+20;-20sin(4x+90)+20`

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