Problem 33 - 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. \(r=3 + 6\ \sin\ \theta)\)

Step-by-Step Solution

Answer

The polar graph is a shifted circle of radius oscillating between 3 and 9 with the symmetry of a \(\sin\) function seen in Cartesian coordinates which translates to symmetry across the polar axis in Polar coordinates.

Step 1: Draw base circle

First thing to note is that \( r = 3 + 6\sin\theta \) is a sinusoidal function modulated by a constant. The constant, 3 in our case, serves as a shift along the radial axis. This indicates that the graph of this equation will be oscillating around the circle of radius 3. Draw this base circle.

Step 2: Identify the 'zeros' and 'maximum r-values'

The zeros of the equation are the points where \( r = 0 \). Solve the equation \( 3 + 6\sin\theta = 0 \) to get the values of \(\theta\) where the equation has zero radius. The maximum values of \( r \) can be found by checking points from the highest value to the constant shift of the sinusoidal function i.e., from 3 to \( 3+6=9 \). Calculate these points and plot them.

Step 3: Utilizing symmetry

Symmetry plays a significant role in polar graphing. In this case, this is a \(\sin\theta \) function, which has symmetry along the y-axis in Cartesian coordinate. In polar coordinate, this translates to symmetry with respect to the polar axis (line \(\theta=0 \)). Plot points mirrored across the polar axis.

Step 4: Plot additional points

To get a complete view of the graph, plot more points on the graph for some other values of \(\theta\). Combine all these points and you should get a clear sketch of the graph.

Key Concepts

Graph Sketching

Understanding graph sketching starts with recognizing the nature of the equation. In our given polar equation, \( r = 3 + 6\sin\theta \), the graph will show sinusoidal behavior due to the term \( 6\sin\theta \). This means the graph will oscillate similar to a wave. However, because of the constant 3 added to \( 6\sin\theta \), the whole graph will be shifted radially outward by 3 units. When sketching polar graphs, always start by identifying these components:

• A base circle, here with radius 3, which serves as a reference.
• The oscillatory part, given by the sinusoidally varying term.
• Key points such as zeros and maximum values which define the shape's outline.

After marking these elements, trace the oscillations of \( \sin\theta \) with the radius shift applied, ensuring to make use of symmetry and additional points to accurately reflect the graph’s behavior.

Symmetry in Graphs

Symmetry is a powerful concept when sketching polar graphs. It helps simplify the drawing process by reducing the amount of data you need to plot manually. In polar coordinates, symmetry can be around the polar axis, the line at \( \theta = 0 \). For our equation, since it's based on \( \sin\theta \), there's inherent symmetry about the polar axis. This mirrors any point across the axis. To take advantage of symmetry:

• Identify symmetric properties related to \( \sin\theta \).
• Mirror plotted points across the polar axis.
• Use these mirrored points to complete the graph, reducing computation and plotting effort.

This ensures that your graph is accurately reflecting the full nature of the polar plot.

Maximum Values

Finding maximum values in a polar graph is crucial because they indicate the furthest radial extent of your graph. In the equation \( r = 3 + 6\sin\theta \), the part \( 6\sin\theta \) reaches a peak of 6 when \( \sin\theta = 1 \), making the maximum value of \( r \) equal to 9 (which is \( 3 + 6 \)). Knowing this max value helps when sketching to see how far the graph stretches radially. To locate maximum values, you should:

• Find when \( \sin\theta \) reaches its peak value, which happens periodically as \( \theta \) changes.
• Calculate \( r \) at these points to identify all radial maxima.
• Mark these on the graph to shape the outer boundaries.

This guides the final shape and ensures you capture the full breadth of the wave-like behavior.

Zeros of a Function

Zeros are pivotal points where the function value \( r \) becomes zero, meaning the polar radius returns to the origin. In our equation \( r = 3 + 6\sin\theta \), zeros occur when \( 3 + 6\sin\theta = 0 \). Solving this gives specific angles where the radius is zero. These zeros reveal entry and exit points of the plot as it crosses the origin. To determine these:

• Solve for \( \theta \) in \( 3 + 6\sin\theta = 0 \), yielding the angles at which the radius is zero.
• Identify all such angles within one period of the sine function, typically 0 to \( 2\pi \).
• Plot these zeros, as they help define the inner crossing points or 'lobes' of the graph.

Understanding and marking these zeros correctly ensures that when you sketch the graph, it includes this central feature of crossing the origin at specific angles.