Question

Graph the polar function on the given interval. \(r=\theta^{2}-(\pi / 2)^{2}, \quad[-\pi, \pi]\)

Short answer

The graph is a parabola rotated in polar coordinates, symmetric around the vertical axis.

Step-by-step solution

Review Polar Coordinates

In polar coordinates, a point is represented by the distance from the origin, denoted as \(r\), and the angle from the positive x-axis, denoted as \(\theta\). Our task is to graph the function \(r = \theta^2 - (\pi/2)^2\) over the interval \([-\pi, \pi]\).

Generate Polar Coordinates

We will calculate \(r\) for a set of \(\theta\) values within the given range \([-\pi, \pi]\). This involves substituting several values of \(\theta\) into the function to get corresponding \(r\) values. For example:- For \(\theta = -\pi\), \(r = (-\pi)^2 - (\pi/2)^2 = \pi^2 - \pi^2/4 = 3\pi^2/4\).- For \(\theta = -\pi/2\), \(r = (-\pi/2)^2 - (\pi/2)^2 = 0\).- For \(\theta = 0\), \(r = 0^2 - (\pi/2)^2 = -\pi^2/4\).- Continue for other values within the range.

Plot Points on Polar Graph

Using the \(r\) values from Step 2, plot each point in the polar coordinates. For each \(\theta\) angle, determine the position from the origin using the computed \(r\) value. Points on the graph will have - Distance \(r\) from the origin,- Angle \(\theta\) measured from the positive x-axis.

Draw the Curve

Connect the plotted points smoothly to illustrate the curve of the function. Since \(r = \theta^2 - (\pi/2)^2\), the graph will be symmetric about the vertical axis (\(\theta=0\)) due to the even nature of the \(\theta^2\) term. Check for symmetry and overlap to ensure the curve is complete.

Key concepts

Polar Graph

A polar graph is a way to visualize mathematical functions using polar coordinates, which differ from the usual Cartesian plane method. Instead of using

• x- and y-coordinates
• polar graphs use a radius, denoted as \( r \)
• and an angle \( \theta \), measured from the positive x-axis.

In these graphs, each point is determined by how far it is from the origin, plus the direction in which it lies based on its angle. This makes polar graphs especially useful for representing equations that are better described in circular paths, spirals, or curves that naturally bend around a center point. By using the polar system, complex relationships between variables can be transparently displayed through unique shapes on a graph. Understanding polar graphs requires a grasp of how circles, angles, and distances interact.

Plotting Polar Functions

Plotting polar functions involves taking a polar equation and translating it into a visual graph. Given a function like \( r = \theta^2 - (\pi/2)^2 \) over the interval \([-\pi, \pi]\), you would follow these steps:

• Start by selecting values of \( \theta \) within your specified range.
• Calculate the corresponding \( r \) values for each selected angle by substituting \( \theta \) back into the equation.
• For example, when \( \theta = -\pi \), the calculation for \( r \) yields \( 3\pi^2/4 \).
• Plot each \((r, \theta)\) pair on the polar coordinate plane.

In this system, each point's distance from the origin depends on its calculated \( r \) value, while its direction hinges on its \( \theta \) angle. When plotted, these points can create intricate and beautiful designs unique to polar functions. To form the complete graph, smoothly connect these plotted points to reveal the continuous nature of the curve.

Symmetry in Polar Graphs

Symmetry plays a crucial role in understanding and creating polar graphs. A polar graph can exhibit different types of symmetry, which helps in predicting the appearance of the graph without needing to plot every point. For the function \( r = \theta^2 - (\pi/2)^2 \), let's explore the common types of symmetry:

• Symmetry about the polar axis (horizontal axis): If for every point \((r, \theta)\), there exists another point \((r, -\theta)\), the graph is symmetric around the horizontal axis.
• Symmetry about the line \( \theta = \pi/2 \) (vertical axis): A graph is symmetric about this axis if, for each point \((r, \theta)\), there is a corresponding point \((r, \pi - \theta)\).
• Symmetry about the origin: This occurs when for every point \((r, \theta)\), another point \((-r, \theta + \pi)\) exists on the graph.

For the given function, the even nature of the \( \theta^2 \) term suggests symmetry about the vertical axis. Recognizing these patterns of symmetry can simplify the graphing process by reducing the number of calculations needed and by visually anticipating the graph's layout.