Problem 31 - 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. \(r=4(1\ +\ \sin\ \theta)\)

Step-by-Step Solution

Answer

The symmetry of \(r=4(1+sin\theta)\) does not lie along the polar axis, the zero of \(r\) is at \(\theta=\pi\), and the maximum \(r\)-value is 8. Additional points can be obtained by choosing various \(\theta\) values across 0 and \(2\pi\) to get a complete graph.

Step 1: Find Symmetry

In order to analyze the symmetry, try substituting \(-\theta\) for \(\theta\) in the equation. If the equation is unchanged, then the graph is symmetrical with respect to the polar axis. After substitution, we get \(r = 4(1+sin(-\theta)) = 4(1-sin\theta)\) which is not same as the original equation, therefore our equation is not symmetric with respect to the polar axis.

Step 2: Determine the Zeros (roots) of the function

Find the value of \(\theta\) which makes \(r=0\). By setting \(r=0\), we get \(\theta = \pi\). This means that the function (\(r\)) is zero at \(\theta = \pi\).

Step 3: Find Maximum \(r\)-values

The maximum value of \(sin\theta\) is 1. Therefore, the maximum \(r\)-value for the given function would be \(r=4(1+1)=8\).

Step 4: Graphing Additional Points

Additional points can be obtained by taking a variety of \(\theta\) values to help give a full image of the graph. Choose a range of \(\theta\) values between 0 and \(2\pi\), add these to the graph and connect the dots along with zeros and maximum \(r\)-values.

Key Concepts

Polar Coordinates

Polar coordinates provide a method for navigating and graphing points on a plane using a radius and an angle, as opposed to the x and y coordinates used in Cartesian systems. To visualize this, think of a polar grid where the origin is called the pole, and from this pole extends a ray called the polar axis (akin to the positive x-axis).

In polar coordinates, a point is defined by \(r,\theta\), where \(r\) is the distance from the pole and \(\theta\) is the angle measured in radians from the polar axis. Positive \(r\) values mean points are in the direction away from the pole, while negative \(r\) values indicate the direction towards the pole. The angle \(\theta\) increases counter-clockwise, which is the positive orientation for angles in polar coordinates.

Symmetry in Polar Graphs

Symmetry in polar graphs simplifies the process of sketching the entire graph, as we can reflect tested portions across lines of symmetry. There are three main types of symmetry to consider in polar graphs:

• Polar axis symmetry: When substituting \(\theta\) with \(\theta\), if the equation remains consistent, the graph has symmetry about the polar axis.
• Origin symmetry: By replacing \(r\) with \(\-r\) and finding the original equation unchanged, the graph is symmetric about the pole (origin).
• Line \(\theta = \frac{\pi}{2}\) symmetry: When we exchange \(\theta\) with \(\pi - \theta\) and if no change is observed in the equation, the graph reveals a reflective symmetry about the vertical line \(\theta = \frac{\pi}{2}\).

Knowing the symmetry of a polar graph helps in understanding its overall structure and it assists in predicting other parts of the graph from the known parts.

Zeros of Polar Functions

The zeros of polar functions are the angle \(\theta\) values at which the radius \(r\) becomes zero. At these angles, the graph crosses the pole (the origin of the polar graph). Identifying the zeros of a polar function is crucial for sketching the graph since it tells us where the graph intersects with the pole.

To find the zeros, we set the polar equation equal to zero and solve for \(\theta\). For the given equation \(r=4(1+\sin\theta)\), by setting \(r=0\), we find \(\theta = \pi\) is zero of the function. This information is pivotal for plotting the initial point from which the graph emerges or where it returns to the pole.

Maximum R-values in Polar Graphs

The maximum \(r\)-values in a polar graph refer to the furthest distance from the pole for given \(\theta\) values. These maxima are often seen as ‘peaks' in the polar graph and occur where the function within the polar equation reaches its own maximum. For instance, in the function \(r=4(1+\sin\theta)\), the maximum value of \(\sin\theta\) can be 1, hence the maximum \(r\) for this function is \(r=4(1+1)=8\).

Finding the maximum \(r\)-values helps in understanding the overall size and scope of the graph. It delineates the outer boundary of the graph, thereby giving a boundary condition for the curve. It's important for plotting the graph accurately, as this point represents the farthest extent of your graph from the pole.