Problem 13 - Precalculus: Functions and Graphs

Question

Find a polar equation that has the same graph as the equation in \(x\) and \(y\). $$x=-3$$

Step-by-Step Solution

Answer

The polar equation is \( r = \frac{-3}{\cos \theta} \).

Step 1: Convert Cartesian Equation to Polar Form

The given equation is in Cartesian form: \( x = -3 \). In polar coordinates, \( x \) and \( y \) are expressed in terms of \( r \) (radius) and \( \theta \) (angle) as: \( x = r \cos \theta \) and \( y = r \sin \theta \). Substitute \( x = r \cos \theta \) into the given equation: \( r \cos \theta = -3 \).

Step 2: Isolate r for the Polar Equation

To express the polar equation, solve for \( r \) in terms of \( \theta \). The equation \( r \cos \theta = -3 \) is already simplified, as \( r \) varies depending on \( \theta \) such that it satisfies the given condition. So, the polar equation is \( r = \frac{-3}{\cos \theta} \), which means \( r \) is determined entirely by \( \cos \theta \).

Step 3: Identify the Graph

The equation \( x = -3 \) represents a vertical line at \( x = -3 \) in Cartesian coordinates. In polar coordinates, \( r = \frac{-3}{\cos \theta} \) describes the same line but expressed in terms relevant to polar graphs. Any \( \theta \) that satisfies this equation will form a line at \( x = -3 \).

Key Concepts

Coordinate Conversion

Coordinate conversion is the process of changing from one coordinate system to another. In mathematics, we often work with Cartesian and polar coordinates. Cartesians involve the typical x and y axes, while polar uses radius r and angle θ. To convert from Cartesian (x, y) to polar (r, θ):

• Use the formula \( x = r \, \cos \, \theta \)
• And \( y = r \, \sin \, \theta \)

The given exercise asks to find the polar form of the equation \(x = -3\). Here, the conversion involves replacing x with \(r \, \cos \, \theta\). Thus, the equation transforms to \(r \, \cos \, \theta = -3\). This translation helps us move effortlessly between different representations of the same geometric entity.

Polar Coordinates

Polar coordinates offer an alternative way of describing points in the plane using a radius and an angle. Let's break it down:

• Radius (r): The distance from the origin to the point.
• Angle (θ): The angle from the positive x-axis to the line connecting the origin to the point.

In our conversion from the exercise, once we solved \(r \, \cos \, \theta = -3\) to get \( r = \frac{-3}{\cos \theta} \), we described the same vertical line equation in polar coordinates. It demonstrates that a vertical line, typically simple in Cartesian form, can be represented using dynamic radial and angular relationships in polar form, dependent on θ.

Cartesian Coordinates

Cartesian coordinates use perpendicular axes to locate a point in a plane. Typically, (x, y) describes a position based on horizontal (x) and vertical distance (y) from a reference point called the origin.
The original exercise's line, \(x = -3\), is a simple vertical line in this system. Every point on this line has an x-value of -3, and y can vary freely.
In Cartesian terms, curves or other shapes are often simpler to graph, especially when compared to polar coordinates, where they require transformations and different variables for representation.

Vertical Line Graph

Graphing vertical lines can help visualize mathematical concepts in a direct way in Cartesian coordinates. A vertical line like \(x = -3\) appears as a straight line parallel to the y-axis intersecting the x-axis at -3. Every point on this line shares the same x-coordinate of -3.
This property simplifies solving many equations as one coordinate remains constant.
In polar coordinates, the same graph requires translating to an equation like \(r = \frac{-3}{\cos \theta}\), as covered in the exercise solution. Remember, graphically both systems describe the same picture, just through a different lens.