Question

Convert the rectangular equation to polar form and sketch its graph. $$ y^{2}=9 x $$

Short answer

The polar form of the equation is \(r\sin^2(\theta) = 9\cos(\theta)\) and the graph is undefined at \(\theta = 0\) with parabolic symmetry.

Step-by-step solution

Conversion of Variables

In polar coordinates, the x and y variables of the rectangular equation can be replaced with \(r\cos(\theta)\) and \(r\sin(\theta)\) respectively. Therefore, the given rectangular equation: \(y^2 = 9x\) can be re-written as \((r\sin(\theta))^2 = 9(r\cos(\theta))\) which simplifies to \(r^2\sin^2(\theta) = 9r\cos(\theta)\).

Simplify the Polar Equation

Rewrite the equation from step 1 by canceling out the common parameter r from both sides giving \(r\sin^2(\theta) = 9\cos(\theta)\). We can only cancel the common r if \(r \neq 0\). This is an essential step to ensure there are no mathematical fallacies in the solution.

Graphing the Polar Function

To graph this function, observe that the function will have values when the right side is equal to the left side. Hence, the equation represents a graph where for every theta, r equals to \(9\cos(\theta)\) divided by \(\sin^2(\theta)\) provided \(\theta \neq 0\). This graph is undefined at \(\theta = 0\), displays symmetry, and will have a parabolic shape.