Problem 54 - Precalculus with Calculus Previews

Question

Suppose \(r=f(\theta)\) is a polar equation. Graphically interpret the given property. $$ f(-\theta)=-f(\theta) \text { (odd function) } $$

Step-by-Step Solution

Answer

The graph is rotated by \(180^{\circ}\) symmetry around the origin.

Step 1: Understanding Polar Coordinate Symmetry

In polar coordinates, a function is symmetric with respect to the origin if, for every point \((r, \theta)\), the point \((-r, \theta)\) is also on the graph. When \(f(-\theta) = -f(\theta)\), it means that the graph of the function has rotation symmetry of \(180^{\circ}\) around the origin.

Step 2: Analyzing the Odd Function Property

The equation \(f(-\theta) = -f(\theta)\) implies that if the point \((r, \theta)\) lies on the graph, then the point \((-r, -\theta)\) also lies on the graph. This is characteristic of a function that is symmetric about both the origin and the polar angle \(\theta = 0\) line.

Step 3: Graphical Interpretation

Graph the points both \((r, \theta)\) and \((-r, -\theta)\). Notice that these two points are reflections of each other across the origin, demonstrating the rotational symmetry. The curve defined by \(f(\theta)\) will appear symmetrical if it is rotated \(180^{\circ}\) around the origin.

Key Concepts

Odd Function

In mathematics, an odd function is one where reversing the sign of the input negates the sign of the output. In formula form, this is written as \( f(-x) = -f(x) \). With polar equations, this concept tweaks slightly since our inputs and outputs are in terms of \( \theta \) and \( r \). For a polar function \( f(\theta) \), being odd implies \( f(-\theta) = -f(\theta) \). For example, if \( f(\theta) \) gives a positive angle and a radius, then \( f(-\theta) \) provides a negative radius for the negative angle. This presents a geometric interpretation where points mirrored across the origin reflect the property of this odd function nature.

Symmetry

Symmetry in polar equations refers to the balanced and mirrored appearance of the graph under certain transformations. One key symmetry is rotational symmetry about the origin. For the polar equation \( r = f(\theta) \), having \( f(-\theta) = -f(\theta) \) indicates rotation symmetry.What this means is:

• Each point \( (r, \theta) \) on the graph has a corresponding point \( (-r, -\theta) \).
• The entire graph looks the same if you turn it 180 degrees around the origin.

This is different from reflection symmetry, typical in Cartesian coordinates. It's helpful to visualize by drawing a few points and noticing where their mirrored counterparts land.

Polar Equation

A polar equation relates a radius \( r \) with an angle \( \theta \). It's written in the form \( r = f(\theta) \), connecting the polar coordinate system's foundations to graphing. You can think of polar equations as a bridge connecting mathematical functions with geometric patterns on a circle. Polar equations are powerful, allowing expressions that are elegant and sometimes simpler than their Cartesian counterparts. Here, symmetry and odd function properties play a pivotal role, giving direct clues to what the graph will look like.For instance, the equation \( r = a \sin(n\theta) \) produces rose-like patterns whose shapes depend heavily on the nature of the angle \( \theta \). Knowing \( f(-\theta) = -f(\theta) \) alerts us that certain shapes on these graphs will be balanced around the origin.

Graphical Interpretation

When interpreting the graph of a polar equation, start by plotting points. For an odd function, these points will demostrate a symmetry as you apply \( f(-\theta) = -f(\theta) \). To see it more clearly:

• Plot the point \( (r, \theta) \).
• Then find its counterpart \( (-r, -\theta) \), which should also be on the graph.
• Continue this for various angles \( \theta \).

This plotting exercise shows how, when you connect these dots, the graph displays a 180-degree rotation symmetry centered at the origin. This means the graph will look identical when rotated half a turn, embodying the properties of an odd function in polar coordinates beautifully. Through this visualization, students gain an intuitive sense of how equations translate into symmetrical shapes.