Problem 71 - Contemporary Precalculus

Question

Sketch the graph of the equation. $$r=2+4 \cos \theta$$

Step-by-Step Solution

Answer

The polar equation $$r = 2 + 4 \cos \theta$$ represents a type of cosine curve in polar coordinates. The graph has symmetry along the polar axis (θ = 0) and a period of $$2\pi$$. The maximum and minimum values of r are in accordance with the cosine function's characteristics.

Step 1: Identify the type of graph

Since the equation involves the cosine function, we can conclude it is a type of cosine curve in polar coordinates. The general equation for such graphs can be represented as: $$r = a \pm b \cos \theta$$ Where 'a' and 'b' are constants. In our case, the equation becomes: $$r = 2 + 4 \cos \theta$$

Step 2: Identify the range of θ

The cosine function is periodic with a period of $$2\pi$$, so we just need to plot the graph for θ in the range $$[0, 2\pi]$$.

Step 3: Analyze the behavior of r as θ changes

We'll generate a table of values for different θ angles, and their corresponding r values. Since cosine has a maximum of 1 and a minimum of -1, we can first analyze those points: - When $$\cos\theta = 1$$, we have $$r = 2 + 4(1) = 6$$. - When $$\cos\theta = -1$$, we have $$r = 2 + 4(-1) = -2$$. Note that a negative value of r means going in the opposite direction, so we actually have a positive r value at an opposite angle. Let's add more points to better understand the curve: - When θ = 0, $$r = 2 + 4(\cos 0) = 6$$. - When $$\theta = \frac{\pi}{4}$$, $$r = 2 + 4\cos(\frac{\pi}{4}) = 2 + 4(\frac{\sqrt{2}}{2}) = 2 + 2\sqrt{2}$$. - When $$\theta = \frac{\pi}{2}$$, $$r = 2 + 4(0) = 2$$. - When $$\theta = \frac{3\pi}{4}$$, $$r = 2 - 4(\frac{\sqrt{2}}{2}) = 2 - 2\sqrt{2}$$.

Step 4: Plot the obtained points and sketch the graph

Plot the points obtained in Step 3 at their respective θ and r values in a polar graph. Then, join these points and continue the curve until you complete one full period of $$2\pi$$ for the angle θ. This should give you a clear understanding of the shape and behavior of the curve.

Step 5: Note the graph's properties and behavior

Observe that the graph has symmetry along the polar axis (θ = 0) and period $$2\pi$$. The maximum and minimum values of r are in accordance with the cosine function's characteristics. Finally, after plotting all the steps, we obtain the graph of the polar equation $$r = 2 + 4 \cos \theta$$.

Key Concepts

Graphing Polar Equations

When we talk about graphing polar equations, we are diving into a unique coordinate system where points are determined by a distance from the origin, called radius \( r \), and an angle \( \theta \) from the positive x-axis. This differs from the usual Cartesian coordinates where points are plotted based on their x and y values. Polar graphs often produce beautiful and complex shapes through these seemingly simple equations.

To graph a polar equation like \( r = 2 + 4 \cos \theta \), start by finding values of \( r \) for different angles \( \theta \). Plot these points on a polar grid: the angle \( \theta \) determines the direction from the origin, and the value \( r \) tells us how far from the origin to plot that point. Join these points smoothly to sketch the graph. Since the cosine function is periodic with period \( 2\pi \), you only need to plot \( \theta \) values from 0 to \( 2\pi \) to see the entire shape.

Use a variety of \( \theta \) values to get accurate curves. Key angles like 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \), as well as others in between, help define the graph’s path.

Cosine Function

The cosine function is central to polar graphs when it forms part of the equation like \( r = 2 + 4 \cos \theta \). In polar coordinates, the cosine function creates symmetrical and periodic patterns due to its properties. Remember that the cosine of an angle will range from -1 to 1.

When you calculate \( r \) for different \( \theta \) values, the behavior of cosine affects your graph. For example:

• At \( \cos \theta = 1 \), the value of \( r \) is at its peak (maximum value).
• At \( \cos \theta = -1 \), \( r \) can even become negative, meaning the point is reflected through the origin.

The important property: cosine's cyclical nature helps predict graph behavior without plotting every point, simplifying the graphing process.

The amplitude and phase shift of the cosine function impact its graph in polar plots. The coefficient before \( \cos \theta \) changes the "outward" reach of the graph, while added constants adjust the graph's base radius.

Polar Graph Symmetry

Symmetry in polar graphs helps in understanding and constructing them effortlessly. For the graph \( r = 2 + 4 \cos \theta \), observe that the graph is symmetric relative to different axes, especially the polar axis.This symmetry means if you graph one section, you can reflect it to fill in the missing parts. Here are some types of symmetry you might see:

• Polar Axis Symmetry : If you replace \( \theta \) with \( -\theta \) and the equation doesn't change.
• Line \( \theta = \frac{\pi}{2} \) Symmetry: Replace \( r \) with \(-r\), the equation remains unchanged. But in this case, the specific symmetry doesn't apply to \( r = 2 + 4 \cos \theta \).
• Origin Symmetry: Replace \( r \) with \(-r\) and \( \theta \) with \( \theta + \pi \), the equation is consistent.

In our equation \( r = 2 + 4 \cos \theta \), the graph retains symmetry along the polar axis, making interpretation and sketching smoother. Recognizing symmetry is beneficial as it reduces the amount of work to graph these beautiful shapes accurately.