Problem 93 - Algebra and Trigonometry

Question

Describe a general procedure for obtaining the graph of \(y=A \sin (B x-C)\)

Step-by-Step Solution

Answer

The graph of \(y=A \sin (Bx - C)\) can be obtained by first identifying \(A\), \(B\), and \(C\) , draw a basic sine graph, apply the amplitude, frequency, and phase shift accordingly.

Step 1: Identify Variables

First, identify the values of \(A\), \(B\), and \(C\) in the given equation. These will determine the characteristics of the graph.

Step 2: Draw the Sine Graph

Plot a basic sine graph on the axes. Ordinarily, this would start at the origin (0,0), extend up to (π/2,1), down to (π,0), continue down to (3π/2,-1), and back up to (2π,0), where it repeats.

Step 3: Apply the Amplitude

Adjust the graph so that it aligns with the 'A' value. The highest point should now be at (\π/2, A\), and the lowest at (\3π/2, -A\).

Step 4: Apply the Frequency

Change the frequency of the graph according to 'B'. If \(B\) is greater than 1, the graph compresses; if \(B\) is less than 1 it stretches horizontally.

Step 5: Apply the Phase Shift

Finally, adjust the graph according to the 'C' value. This will move the graph to left or right. If \(C\) is positive, the graph shifts to the right, and if \(C\) is negative, it shifts to the left.