Question

Sketch a graph of the polar equation and identify any symmetry. $$ r=2 \theta $$

Short answer

The graph of \( r = 2\theta \) is a spiral without symmetry.

Step-by-step solution

Understand the Polar Equation

The given polar equation is \( r = 2\theta \). This means that the distance from the origin, \( r \), is twice the angle \( \theta \). The angle \( \theta \) is usually measured in radians.

Create a Table of Values

Choose several values of \( \theta \) and calculate the corresponding \( r \). For example, when \( \theta = 0 \), \( r = 0 \); when \( \theta = \pi/4 \), \( r = \pi/2 \); when \( \theta = \pi/2 \), \( r = \pi \), and so forth.

Plot Points on the Polar Grid

Plot the points from the table on the polar grid. Remember that each point is a combination of \( r \) and \( \theta \). For \( \theta = 0 \), the point is at the origin; for \( \theta = \pi/4 \), plot a point at \( \pi/2 \) units along the line at \( \pi/4 \) radians, and so on.

Draw the Graph

Connect the plotted points to form the graph of \( r = 2\theta \). Because \( \theta \) increases linearly, this graph will form a spiral. As \( \theta \) increases, \( r \) also increases, unwinding from the origin.

Analyze Symmetry

Determine if the graph has any symmetry. In polar coordinates, check for symmetry with respect to the polar axis, the line \( \theta = \pi/2 \), and the pole. In this case, \( r = 2\theta \) does not show symmetry with respect to any of these lines or points, as it spirals out without mirroring across any axes.

Key concepts

Polar Equation

A polar equation is a mathematical expression that describes a curve using polar coordinates. In contrast to Cartesian coordinates, where we use the

• x-axis
• y-axis

polar coordinates utilize distances and angles. The distance from a fixed point (usually the origin) is denoted as \( r \), and the angle from a fixed direction (usually the positive x-axis) is \( \theta \).
For the equation \( r = 2\theta \), \( r \) depends on the angle \( \theta \). This implies that as the angle changes, the radius or distance from the origin changes too. The concept of a polar equation is crucial for plotting many shapes, including spirals and circles.
Visualizing polar equations can be easier once you remember that every point is given by its distance and angle. Instead of moving up and down or left and right, like in a Cartesian system, you rotate and stretch outwards to locate each point.

Symmetry

Symmetry in polar equations is about whether the graph looks the same across a line or point. In polar coordinates, symmetry can be observed with respect to:

• the polar axis, which is the equivalent of the x-axis in Cartesian coordinates
• the line \( \theta = \pi/2 \), which is the same as the y-axis
• the pole, which is the origin point

Checking for symmetry involves examining if the graph reflects about these lines or points. For the equation \( r = 2\theta \), the graph is a spiral extending outward.
This spiral does not mirror or repeat across the polar axis, the line \( \theta = \pi/2 \), or the pole, since it continuously winds around without doubling back. Therefore, \( r = 2\theta \) has no symmetry in the traditional sense of these axes.

Graph of Spiral

The graph of a spiral in polar coordinates begins with an equation like \( r = 2\theta \). This type of spiral is known as an Archimedean spiral, where the distance between successive turns is constant.
To sketch this spiral, consider a few points:

• Start with \( \theta = 0 \) resulting in \( r = 0 \) at the origin.
• Move to \( \theta = \pi/4 \) to find \( r = \pi/2 \).
• Continue this pattern, plotting points, as \( \theta \) and \( r \) increase.

These calculated points are placed on a polar grid, which is a circular graph paper where each concentric circle equates to different distances \( r \).
As more points are plotted and connected, a spiral starts to form. This spiral consistently moves away from the origin as \( \theta \) increases. Its appearance resembles a coil unwinding with each step, maintaining a steady expansion outward. Understanding this pattern helps visualize complex curve formations created by polar equations.