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Chapter 4: Problem 112

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

Short Answer

Expert verified
The statement can make sense if you understand the cosine function and find it easier to start graphing from the x-axis crossings. However, this is largely dependent on personal preference and the specific function being graphed.

Step by step solution

01

Understanding the cosine function

The cosine function \(y=A \cos (B x-C)\) has a general shape that depends on the values of \(A\), \(B\), and \(C\). The number \(A\) determines the amplitude or height of the wave, \(B\) determines the period or width of one complete cycle, and \(C\) shifts the graph horizontally. A complete cycle starts and ends at the same position, and includes all the peak, trough and zero-crossing points in between.
02

Starting the graph

When you start graphing at the \(x\)-axis, you're actually starting from the point where the graph crosses the \(x\)-axis. For the cosine function, these occur at \(x = C/B\) and \(x = 2蟺/B + C/B\) within one complete cycle. This could make graphing easier because the cosine function has a known value at these points (typically 1 or -1), which provides a clear starting point.
03

Considering the statement

If you understand the shape of the cosine function and are comfortable starting the graph at the \(x\)-axis crossings, then the statement can make sense. It provides a known, fixed point to start from. However, given that the cosine function is periodic and its graph would look the same wherever you start from, this statement is largely dependent on personal preference and the specific function you're graphing.

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