Problem 43 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=2 \sec \theta$$

Step-by-Step Solution

Answer

The graph of \(r = 2 \sec(\theta)\) consists of a series of vertical, open-ended curves intersecting the axis at \(\theta = n\pi\) with asymptotes between these points. The graph does not possess symmetry about the polar axis.

Step 1: Identify Zeroes and Asymptotes

To find the zeroes of the function, solve the equation \(r = 0\), which yields \(\theta = n\pi\), where \(n\) is an integer. For the asymptotes, solve \(2 \sec(\theta) = \infty\), which yields \(\theta = (2n+1)\frac{\pi}{2}\), where \(n\) is an integer.

Step 2: Identify Symmetry

To check if the equation possesses symmetry, apply the rules. Replacing \(\theta\) by \(-\theta\) and \(r\) by \(-r\), and if it simplifies back to the original equation, the equation has symmetry about the polar axis. However, in this case, replacing does not yield back the original equation. Hence, the graph does not possess symmetry about the polar axis.

Step 3: Plot the Graph

Using all the information gathered in previous steps, sketch the graph. Draw a horizontal axis representing \(\theta\) and a vertical axis representing \(r\). Mark the values of \(\theta\) at which \(r = 0\), draw vertical asymptotes at \(\theta = (2n+1)\frac{\pi}{2}\), and finally sketch the graph of \(r = 2 \sec(\theta)\). The resulting graph will consist of a series of vertical, open-ended curves intersecting the axis at \(\theta = n\pi\) with asymptotes between these points.

Key Concepts

Symmetry in Polar Coordinates

Symmetry in polar coordinates is an essential concept used to simplify graphing polar equations. When graphing in polar coordinates, symmetry can tell you if a shape can be reflected over an axis or a point. There are three main types of symmetry in polar graphs:

• Polar Axis Symmetry: The graph looks the same above and below the polar axis. Replacing \(\theta\) with \(-\theta\) should return the original equation.
• Line \(\theta = \frac{\pi}{2}\): The graph looks the same to the left and right of this line. Replacing \((r, \theta)\) by \((-r, -\theta)\) should give an equivalent equation.
• Origin Symmetry: The graph can be rotated by \(\pi\) and still looks the same. Here, replace \((r, \theta)\) with \((-r, \theta+\pi)\) or \((r, \theta+\pi)\).

In our example equation \(r = 2 \sec \theta\), testing for symmetry shows that it has no symmetry about the polar axis or origin. These tests help understand and predict the shape of the graph, though in this case, they do not apply.

Zeros of Polar Functions

Zeros represent the angles \(\theta\) at which the radial coordinate \(r\) equals zero. Finding zeros is critical because it tells us where the graph will touch or cross the pole. For a given polar equation like \(r = 2 \sec \theta\), set \(r = 0\).
By solving, you find \(\theta = n\pi\), where \(n\) is an integer.These zeros indicate positions where the function meets the origin. This information helps sketch the graph accurately by showing intersections with the origin.

Asymptotes in Polar Graphs

Asymptotes in polar graphs are lines the curve approaches but never crosses. They help depict the graph's behavior at extreme values. To determine asymptotes for \(r = 2 \sec \theta\), observe where the function becomes undefined.
Since \(\sec \theta = \frac{1}{\cos \theta}\), it will be undefined when \(\cos \theta = 0\).
Solving \(\cos \theta = 0\) provides \(\theta = \frac{(2n+1)\pi}{2}\), marking vertical asymptotes.These values cause the function to "blow up," indicating where the graph extends infinitely. Marking asymptotes helps structure graphs by creating barriers and showing long-tail behavior.

Graphing Polar Equations

Graphing polar equations involves using the information collected about symmetry, zeros, and asymptotes.
In our case, \(r = 2 \sec \theta\) translates into graphing distinct curves around asymptotes and emphasizing zeros.

• First, sketch axes, with angles \(\theta\) on the horizontal and radius \(r\) on the vertical.
• Next, mark zeros at \(\theta = n\pi\).
• Draw vertical asymptotes at \(\theta = \frac{(2n+1)\pi}{2}\).
• Finally, sketch the curve, respecting zeros and asymptotes placement.

No symmetry simplifies the process, so take care to ensure vertical curves pass the symmetry test visually. This comprehensive understanding simplifies plotting complex graphs efficiently, even for those first learning polar equations.