This is a tutorial on the relationship between the amplitude, the vertical shift and the maximum and minimum of the sine function.
Problem 1: Function F is a sine function given by
F(x) = a*sin(bx + c) + d
with a > 0.
 Show that the maximum value Fmax and the minimum value Fmin of F(x) are given by
Fmax = d + a
Fmin = d  a
 Show that d = (Fmax + Fmin) / 2
 Show that a = (Fmax  Fmin) / 2
Solution to Problem 1:
 We start by using the fact that
1 <= sin(bx + c) <= 1
 Multiply all terms of the above double inequality by a
a <= sin(bx + c) <= a
 Add d to all terms of the double inequality above
d  a <= sin(bx + c) + d <= d + a
 sin(bx + c) + d is the expression that define F(x), hence
d  a <= F(x) <= d + a
 F(x) has a minimum value Fmin and a maximum value Fmax given by
Fmin = d  a
Fmax = d + a
 Add the left terms and the right terms of
Fmax = d + a and Fmin = d  a to obtain
Fmax + Fmin = 2d
 Divide both sides of the above by 2 to obtain
d = (Fmax + Fmin) / 2
 Add the left terms and the right terms of
Fmax = d + a and Fmin = d + a to obtain
Fmax  Fmin = 2a
 Divide both sides by 2 to obtain
a = (Fmax  Fmin) / 2
Problem 2:The graph of the sine function F given by
F(x) = a*sin(x) + d
with a > 0, is shown below.
Use the graph and the results of problem 1 above to find a and d.
Solution to Problem 2:
 From the graph, the maximum value Fmax = 6 and the minimum value Fmin = 2, hence
d = (Fmax + Fmin) / 2 = 2
and
a = (Fmax  Fmin) / 2 = 4
More references on
Trigonometry Problems.
Match Sine Functions to Graphs
