Tutorial on Sine Functions (2)- Problems

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.
  1. Show that the maximum value Fmax and the minimum value Fmin of F(x) are given by
    Fmax = d + a
    Fmin = d - a
  2. Show that d = (Fmax + Fmin) / 2
  3. Show that a = (Fmax - Fmin) / 2

Solution to Problem 1:

  1. We start by using the fact that
    -1 <= sin(bx + c) <= 1
  2. Multiply all terms of the above double inequality by a
    -a <= sin(bx + c) <= a
  3. Add d to all terms of the double inequality above
    d - a <= sin(bx + c) + d <= d + a
  4. sin(bx + c) + d is the expression that define F(x), hence
    d - a <= F(x) <= d + a
  5. F(x) has a minimum value Fmin and a maximum value Fmax given by
    Fmin = d - a
    Fmax = d + a
  6. Add the left terms and the right terms of
    Fmax = d + a and Fmin = d - a to obtain

    Fmax + Fmin = 2d
  7. Divide both sides of the above by 2 to obtain
    d = (Fmax + Fmin) / 2
  8. Add the left terms and the right terms of
    Fmax = d + a and -Fmin = -d + a to obtain

    Fmax - Fmin = 2a
  9. 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.
Graph of Sine Function

Solution to Problem 2:

  1. 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 and Links

Trigonometry Problems.
Match Sine Functions to Graphs

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