Problem 92 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm graphing a polar equation in which for every value of \(\theta\) there is exactly one corresponding value of \(r,\) yet my polar coordinate graph fails the vertical line for functions.

Step-by-Step Solution

Answer

The statement does not make sense. If for each \( \theta \) there is exactly one corresponding \( r \), then it's likely the graph would pass the vertical line test in a polar coordinate system if graphed correctly. The observed failure might be due to incorrect plotting or a misunderstanding of vertical line test application in polar coordinates.

Step 1: Understanding the Statements

The first part of the statement is about polar equations having only one value for \( r \) for every \( \theta \). Now, this would typically imply that, in the polar coordinate system, for each angle, there is only one distance from the origin, rendering a function-like setup. However, the second part of the statement contradicts this, proclaiming failure of the vertical line test which is a method to determine if a graph is a function or not. Any graph that passes this test represents a function.

Step 2: Processing the Contradiction

The contradiction between the two parts of the statement hints at a misunderstanding. If for every \( \theta \) there is exactly one corresponding \( r \), that should mean the graph can pass the vertical line test since it signifies a function-like behavior.

Step 3: Final Conclusion

Given the above analysis, it seems there might be a mistake in graphing the polar coordinate. If the coordinates are correct, then the only reason the graph would fail the vertical line test might because either it is not properly plotted, or a misunderstanding of how the vertical line test applies to polar coordinates (a vertical line can mean constant \( r \) instead of constant \( \theta \)).