Question

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{1}{1-2 \cos \theta}$$

Short answer

a) \(\frac{x}{\cos \theta} = \frac{1}{1-2 \cos \theta}\) b) \(\frac{y}{\sin \theta} = \frac{1}{1-2 \cos \theta}\) c) \(y = \frac{x \sin \theta}{1 - 2 \cos \theta}\) d) \(x = \frac{y \cos \theta}{1 - 2 \sin \theta}\)

Step-by-step solution

Convert polar equation to Cartesian equation

To convert the polar equation to a Cartesian equation, we will use the following conversions: $$x = r \cos \theta \Rightarrow r = \frac{x}{\cos \theta}$$ $$y = r \sin \theta$$ Now let's substitute these conversions into the polar equation: $$ \frac{x}{\cos \theta} = \frac{1}{1-2 \cos \theta}$$ We can now solve for \(y\): $$y = \frac{x \sin \theta}{1 - 2 \cos \theta}$$

Plot the equation on the Cartesian plane

Now that we have the Cartesian equation, we can plot it on the Cartesian plane. We can use a graphing calculator or software to generate the graph, or we can plot \(\theta\) values in increments from \(0\) to \(2\pi\) and connect the points. Note that you should pay attention to the behavior of the graph when \(\theta\) is close to the values where the denominator becomes zero (\(\cos \theta = \frac{1}{2}\)), as the graph approaches asymptotes in these regions.

Label points and indicate curve's generation using arrows

Once the graph is plotted, we can label various points on the curve with their corresponding \(\theta\) values, to give an idea of the direction in which the curve is generated. Be sure to indicate points where \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3 \pi}{2}\), and \(2 \pi\). Additionally, use arrows to show the direction the curve is being generated as \(\theta\) increases from \(0\) to \(2 \pi\). With these steps, the graph of the polar equation \(r=\frac{1}{1-2 \cos \theta}\) is complete, showing how the curve is generated as \(\theta\) increases from \(0\) to \(2 \pi\).

Key concepts

Polar to Cartesian Conversion

Converting polar equations to Cartesian coordinates can be quite challenging, but understanding the relationship between these two systems is key. In the context of the provided exercise, to transform the polar equation \( r=\frac{1}{1-2 \cos \theta} \) to its Cartesian form, one uses the foundational conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \).

When substituting these into the original polar equation, it’s crucial to treat \( r \) as the hypotenuse of a right-angled triangle, with \( x \) and \( y \) representing the adjacent and opposite sides respectively, depending upon the angle \( \theta \). As one substitutes these relationships, algebraic manipulation allows to express \( y \) as a function of \( x \), creating a Cartesian equation which can then be graphed on the traditional x-y coordinate plane. This process provides insight into the shape of the curve as well as the location of key features such as intercepts or asymptotes.

Asymptotes in Polar Graphs

In the realm of polar graphs, an asymptote may not be as straightforward to identify as with Cartesian graphs. However, understanding their occurrence in polar equations is essential. Asymptotes in polar graphs occur when the radius \( r \) stretches infinitely in a certain direction, which happens when the denominator of an equation approaches zero.

In the exercise \( r=\frac{1}{1-2 \cos \theta} \), the asymptotes appear when \( 1-2 \cos \theta = 0 \), that is, when \( \cos \theta = \frac{1}{2} \). This happens at specific values of \( \theta \), leading to a division by zero in the polar equation. Therefore, it's critical to keep an eye on these trouble spots where the value of \( \theta \) makes the cosine function equal to 1/2, as the radius will grow without bounds, creating an asymptote visible on the graph.

Plotting Polar Coordinates

Plotting polar coordinates is a process distinctly different from Cartesian plotting. It involves marking the distance from the origin (radius, \( r \)) and the angle from the positive x-axis (\( \theta \)), typically in radians. In working through our exercise, one must consider various values of \( \theta \), ranging from 0 to \( 2\pi \), to trace the curve's path.

To visualize the equation, one plots points for selected angles and connects these to illustrate the curve. Arrows are then used to show the curve’s direction of generation as \( \theta \) increases, providing clarity on how the graph evolves over a complete cycle. Labeling significant points such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \) spills light on the graph’s symmetry and periodicity, pivotal attributes of polar graphs.