Problem 111 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing one complete cycle of \(y=A \sin (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Step-by-Step Solution

Answer

The statement makes sense only when there is no phase shift in the sine function, i.e., when C = 0. If C ≠ 0, the graph wouldn't necessarily start on the x-axis. Hence, it's not a universal rule for all sine functions.

Step 1: Understanding the basics of a sine function

The general form of a sine function is \(y=A \sin (Bx-C)\), where A represents the amplitude, B affects the period of the function and C affects the phase shift. The sine function has a range from -A to A, a period of \(2\pi/B\), and it is displaced C units horizontally. A complete cycle of the sine curve starts and ends at the same point; hence the curve repeats itself after every complete cycle.

Step 2: Analyzing the statement

The statement suggests that the graph of \(y=A \sin (Bx-C)\) should begin at the x-axis. This would be true if there were no phase shift, i.e., if C=0. In that case, the graph does indeed start from the x-axis. However, if there is a phase shift (C ≠ 0), the graph wouldn't necessarily begin on the x-axis. Hence the statement makes sense under certain conditions, but it's not a universal rule for all sine functions.

Step 3: Final evaluation

Finally, looking at the above analysis, it's clear that the statement 'When graphing one complete cycle of \(y=A \sin (Bx-C)\) I find it easiest to begin my graph on the x-axis.' only makes sense when there is no phase shift (i.e., C = 0). If there is a phase shift, the starting point of the graph on the x-axis would be shifted by C units. So, it's not a universal rule and depends on the values of the function parameters.