Write a cosine functionfx such that on one period a local ma

Write a cosine function.f(x), such that on one period a local maximum is at (-pi/3,1)) followed by a local minimum at (2pi/3,-7).

The vertical distance between maximum and minimum value is obtained by finding subtraction of y cordinates:

1-(-7) = 8

Now 8/2 = 4 is amplitude of our function

Now going down from the maximum or (up from the minimum) by the amount of amplitude 4 will give the vertical shift of the function

Hence 1-4 = -3 is vertical shift

The period is twice the horizontal distance between two consecutive maximum and minimum points.

The period = 2(2pi/3-(-pi/3)) = 2(3pi/3) = 2pi

Now our base function is y=cosx

Here y is at maximum value when x=0.The phase shift for a cos function is how far from 0 x is when the first maximum occurs.

And we have maximum at -pi/3 , Hence phase shift is -pi/3

Y = Acos(B(x-D)) + C

Here C is the veryical shift and so C = -3

B is the period number = 2pi/period = 2pi/2pi = 1

A is amplitude = 4

D is phase shift and hence -pi/3

The function is Y = 4cos(x+pi/3)-3

2) The amplitude is given by (4-(-2))/2 = 6/2 = 3

Going up 3 units from -2 will give vertical shift; Vertical shift = -2+3 = 1

Now period = 2(pi/2-(-pi/2)) = 2(2pi/2) = 2pi

So period number = pi/period = pi/2pi = 1/2

Phase shift is No phase shift

Hence we get f(x) = 3tan(1/2(x))+1 = 3tan(x/2) + 1