Problem 38 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=4 + 3 \cos\ \theta\)

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 4 + 3 \cos\ \theta\) is a circle with a center moved from the origin (0,0) to point (4,0) and with a radius of 3. It has symmetry around the vertical line, but has no real zero. The maximum \(r\)-value is 7 at angles 0 degree and 360 degrees.

Step 1: Evaluate the graph at key points,

Before plotting the graph, it's important to understand some crucial points. Since this is a cosine function, evaluate the graph at \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\). For instance, the points of the polar equation are: (4,0), (7, \(\frac{\pi}{2}\)), (4, \(\pi\)), (1, \(\frac{3\pi}{2}\)), and back to the start point at (4,0).

Step 2: Sketch the graph of the polar equation using symmetry,

This graph has symmetry, meaning that opposite points draw the same values. If you draw a line through the origin and points at \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\), you should note that the distance from the origin is the same from both points to the origin, telling you the graph is symmetric.

Step 3: Find the zeros

The zero of our polar equation is the value of \(\theta\) that makes \(r = 0\). In this polar equation, \(r = 0\) when \(\cos(\theta) = -4/3\), which doesn't correspond to the given \(\theta\)-values we can get from the unit circle. So, there is no real zero for this set of polar coordinates.

Step 4: Find the maximum \(r\)-values

The maximum \(r\)-value is found when the \(\cos\theta\) is maximum. The maximum value of \(\cos\theta\) is 1. So, the maximum \(r\)-value of our polar equation occurs at \(\theta = 0\) and \(\theta = 2\pi\) where \(r = 4 + 3 * 1 = 7\). So the maximum \(r\)-value of our polar equation is 7.

Step 5: Sketch the graph of the polar equation

Begin at the pole (the origin in polar coordinates). Start from (4, 0) and you'll see that the graph appears like a circle that moved its center from the origin (0,0) to point (4,0) with a radius of 3.

Key Concepts

Symmetry in Polar Graphs

Symmetry in polar graphs simplifies the sketching of these equations by allowing us to predict and reflect points about axes or the pole. In polar coordinates, symmetry can occur in three main forms:

• Symmetry about the Polar Axis: If replacing \(\theta\) with \(-\theta\) gives the same \(r\)-value, the graph is symmetric about the polar axis (horizontal line).
• Symmetry about the Line \(\theta = \frac{\pi}{2}\): If replacing \(\theta\) with \(\pi - \theta\) yields the same \(r\)-value, the graph is symmetric around this vertical line.
• Symmetry about the Pole: If \(r\) becomes \(-r\) for some \(\theta\), then the graph is symmetric about the pole (the origin).

In the case of the equation \(r = 4 + 3 \cos \theta\), symmetry is useful. By testing angles \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), you observe the function's reflection over these points, ensuring that the graph looks the same on either side of the line \(\theta = \frac{\pi}{2}\). This type of symmetry assists in accurate and efficient graph sketching.

Maximum r-value

The concept of the maximum \(r\)-value in polar coordinates denotes the furthest point reached from the origin. To find the maximum \(r\)-value, we should determine when the trigonometric component of the polar equation reaches its peak. For equations involving cosine, such as \(r = 4 + 3 \cos \theta\), the maximum value of \(\cos \theta\) is 1. Substituting this into the equation gives the largest \(r\)-value.

For example, when \(\cos \theta = 1\) (at \(\theta = 0\) and \(2\pi\)), our polar equation becomes \(r = 4 + 3 \cdot 1 = 7\). Thus, the point furthest from the origin is at \(r = 7\). Understanding the maximum \(r\)-value helps us determine the boundaries of the graph and focus on key features like size and direction.

Sketching Polar Equations

Sketching polar equations requires insight into the equation's behavior through its key points, symmetry, and maximum \(r\)-values. A good sketching technique begins with evaluating the function at strategic angles to get a sense of the graph's overall shape.

Steps to sketch a polar equation like \(r = 4 + 3 \cos \theta\):

• Evaluate the equation at fundamental angles \(\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and \(2\pi\), and plot these points.
• Use symmetry to reflect points across the lines mentioned in the symmetry section.
• Identify the maximum \(r\)-value at \(\theta = 0\) or \(2\pi\) to emphasize these on your sketch.
• Plot additional points if needed for illustration of curvature and check if the graph encloses certain symmetrical shapes, such as circles or ellipses.

The graph for \(r = 4 + 3 \cos \theta\) resembles a circle slightly shifted along the polar axis. By incorporating these steps, you'll create an accurate representation of the polar graph.