Problem 58 - Precalculus

Question

Graph the polar equation. \(r=-3 \cos (4 \theta)\)

Step-by-Step Solution

Answer

The graph is a rose curve with 8 petals, each with a length of 3 units.

Step 1: Identify the Type of Polar Equation

The given polar equation is of the form \(r = a \cos(n\theta)\), where \(a = -3\) and \(n = 4\). This is the equation for a rose curve, which is symmetric about the x-axis when the cosine function is used.

Step 2: Determine Number of Petals

For a rose curve \(r = a \cos(n\theta)\), if \(n\) is even, the curve will have \(2n\) petals. In this case, \(n=4\), so the graph will have \(8\) petals.

Step 3: Calculate Maximum Length of Petals

The maximum length of each petal is given by the absolute value of \(a\). Here, \(a = -3\), so the length of each petal is 3 units.

Step 4: Plot Key Points and Graph the Curve

To plot the rose curve, we need to evaluate \(r\) for several angles \(\theta\) to identify where the petals are located. Due to symmetry and periodicity, it is efficient to first plot values within one full rotation of \(\theta\) (from 0 to \(\pi\) since \(n\) is even). This helps in visualizing the repeated pattern of petals.

Step 5: Visualize the Graph

Draw the circle plot using polar coordinates. Since there are 8 petals, they are distributed evenly around the circle. There will be four petals along the coordinate axes, matched with their negative counterparts due to the cosine factor being negative.

Key Concepts

Rose Curve

The concept of a rose curve is central to understanding certain polar equations. A rose curve is a type of polar graph characterized by its petal-like appearance. It is formed when the equation is in the form \(r = a \cos(n\theta)\) or \(r = a \sin(n\theta)\). The variable \(n\) determines the number of petals the curve will have. If \(n\) is even, the rose curve has \(2n\) petals. Conversely, if \(n\) is odd, it will have \(n\) petals. The value of \(a\) affects the amplitude, determining the length of each petal. In our example, since the equation is \(r = -3 \cos(4\theta)\), it is a rose curve with 8 petals, as \(n\) is 4 and thus even.

Symmetry in Polar Graphs

Symmetry plays an important role in analyzing polar graphs. For polar equations involving cosine functions, such as \(r = a \cos(n\theta)\), the graph is symmetric about the x-axis. This implies that if a part of the graph extends above the x-axis, there will be a corresponding part below it, mirrored across the x-axis. This symmetry simplifies plotting and visualizing the rose curve since you only need to compute key points for half the polar plane to predict the entire pattern. This inherent symmetry helps simplify calculations and graphing processes.

Graphing Polar Coordinates

Graphing polar coordinates requires a unique approach compared to its Cartesian coordinate counterpart. When dealing with polar equations like \(r = -3 \cos(4\theta)\), one must carefully choose values for \(\theta\) and compute the corresponding \(r\). The pair \((r, \theta)\) helps in plotting points on a circular grid rather than a rectangular one. Since the radius \(r\) can take negative values, it indicates a direction opposite to that given by \(\theta\). This distinction is crucial in constructing plots accurately, especially for curves like the rose curve which exhibit radial symmetry.

Cosine Function in Polar Equations

The cosine function significantly influences the design and shape of polar equations. In polar form, \(r = a \cos(n\theta)\), cosine determines how the graph behaves about the x-axis. Since cosine is an even function, it naturally leads to symmetry, often resulting in even distributions like the petals in a rose curve. Moreover, when a negative constant multiplies the cosine function, such as \(-3\) in our example, it inverts or reflects the graph about the x-axis. The negative value of \(a\) flips the petals to the opposite side, adding another layer of symmetry and visual interest to the graph.