Question

Explain why the vertical-line test used to identify functions

in rectangular coordinates does not work for equations

expressed in polar coordinates.

Short answer

The reason is explained in subsequent steps.

Step-by-step solution

Step 1. Given

Vertical line test and rectangular cordinates.

Step 2. Explanation

Here we have to give reason, why the vertical line test used to identify functions in rectangular

coordinates does not work for equations expressed in polar coordinates.

Vertical line test in rectangular coordinates is used to get symmetry of an equation or function

along vertical y axis, in which we replace x coordinate of any point on the graph, with –x, so that

to get its left or right image along y axis.

Now as we have that polar coordinate is represented as $\left(r,\theta \right)$, where r is the distance of the

point from the pole of polar axis. Now on applying the same ruling of rectangular coordinates, in

case of polar coordinates also, or on replacing r by –r, we find another point on the same terminal

line at the same distance r from pole, but it different direction, which means that on such

replacement, symmetry lies along the pole, not along the vertical line as it lies in the case of

rectangular coordinates.