Problem 112 - 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 \(y=A \cos (B x-C)\) I find it easiest to begin my graph on the \(x\) -axis.

Step-by-Step Solution

Answer

The statement does not universally make sense. The starting point of the graph depends on the function parameters A, B, and C. Without specific values for A, B, and C, we cannot confirm that starting on the \(x\)-axis is always the easiest method.

Step 1: Understand Characteristics of Cosine Function

The standard form of a cosine function is \(y=A \cos (B x-C)\) where:\n-A is the amplitude.\n-B determines the period of the function. The period of the function is \(\frac{2\pi}{|B|}\).\n-C is the horizontal shift also known as the phase shift. If C is positive, the graph moves to the right; if C is negative, the graph moves to the left.

Step 2: Analyze the Statement

The statement expresses that it is easiest to begin graphing the function from the \(x\)-axis. As the standard cosine function \(y= \cos(x)\) starts from a maximum value (when \(x=0\), \(y=1\)) and the parameter A, B, and C transformations could affect the starting point, the claim to always start the graph from the \(x\)-axis is not accurate. However, without specific values for A, B and C known, further confirmation cannot be made.

Step 3: Conclusion

Based on the nature of the cosine function and the impact of parameters A, B, and C on the function, it cannot be definitively stated that starting the graph from the \(x\)-axis is the easiest or most effective method for all cases. Each graph of the cosine function needs to be approached based on its individual parameters.