Question

Find the equations of the form f(x)=a*cos(bx+c)+d, with a maximum point (-pi /2,5) and a minimum point (pi /4,-4)

Answer 1

amplitude (a) is (max-min)/2
center line (d) is (max+min)/s
cos(bx) has period 2π/b. So, assuming that the given max and min are the closest together, they span 1/2 period.

Once you have b, cos(bx+c) = cos(b(x + c/b))
cosx has a max at x=0, so to find c, you need x + c/b = 0

Answer 2

distance between max and min = 1/2 of a period
pi/4 - (-pi/2) = 3pi/4
2pi/k = 6pi/4
2/k = 3/2
k = 4/3
range from max to min = 5-(-4) = 9
a = 4.5
d = 0.5
sofar f(x) = 4.5 cos (4x/3 + c) + 0.5
when x = pi/4, y = 4
4 = 4.5cos(pi/3 + c) + 0.5
cos (pi/3 + c) = 7/9
pi/3 + c = 6796738...
c = -.3675...

so f(x) = 4.5 cos (4x/3 - 3675) + 0.5
check my arithmetic, I should have written it out first

Answer 3

4/3 (x + 3c'/4) is a max at -π/2 + 3c/4 = 0
c' = π/3
so, g(x) = 9/2 cos(4/3 (x + 2π/3)) + 1/2 = 9/2 cos(4/3 x + 8π/9)