Question

a cosine function that passes through the points (1/2,1) and (3/2,-3).

Maximum value: y=1
Minimum value: y=-3
Midline: y=-1

determine b

Answer 1

Given the points (1/2, 1) and (3/2, -3), we can use the cosine function equation:

y = a cos(bx) + c

Substitute the points into the equation:

1 = a cos(b(1/2)) + c
-3 = a cos(b(3/2)) + c

This gives us the two equations:

1 = a cos(b/2) + c
-3 = a cos(3b/2) + c

Solving for c in the first equation:

c = 1 - a cos(b/2)

Substitute c into the second equation and solve for a:

-3 = a cos(3b/2) + 1 - a cos(b/2)
-4 = a (cos(3b/2) - cos(b/2))
a = -4 / (cos(3b/2) - cos(b/2))

Now that we have the values of a and c, we can determine the value of b using the midpoint of the maximum and minimum values of the function:

Midpoint = (Maximum + Minimum) / 2
-1 = (1 + (-3)) / 2

Substitute into the original equation:

-1 = a cos(bx) + c
-1 = a cos(bx) + 1 - a cos(b/2)
-2 = a (cos(bx) - cos(b/2))

Solve for b:

-2 = a (cos(bx) - cos(b/2))
-2 = -4 / (cos(3b/2) - cos(b/2)) (cos(bx) - cos(b/2))
cos(bx) - cos(b/2) = 2 / (cos(3b/2) - cos(b/2))

This will give us the value of b for the cosine function that passes through the given points.