Expert: Ahmed Salami - 9/14/2009

Question
On a graph of y = cos(Ax) + B, points P and Q are consecutive lowest and highest
points. Find the slope of the line through P and Q.  

Answer
Hi Abdullah,
The graph of y = cos(Ax) + B is simply the graph of y = cos(Ax) moved up B units and has a period of 2π/A.
The turning points occur when dy/dx = 0 i.e sin(Ax) = 0 which gives us Ax = 0,π,2π,.....
Or simply x = 0,π/A,2π/A,......
Starting from x = 0 which is a 'highest' point the other points alternate between lowest and highest values. Taking the next two points x = π/A and 2π/A which are consecutive lowest and highest points, the values of y at these points are
y = cos[A(π/A)] + B and cos[A(2π/A)] + B
y = cos(π) + B and cos(2π) + B
y = -1 + B  and  1 + B
The slope is therefore [(1 + B) - (-1 + B)]/[(2π/A) - (π/A)]
= 2/(π/A)
= 2A/π
This is somewhat a rigorous explanation, its quite easier to just figure out.

Regards