Problem 92 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=-4 \cos ^{2}(\theta / 2)$$

Step-by-Step Solution

Answer

The graph resembles an origin-symmetric floral pattern.

Step 1: Understand the Polar Equation

First, interpret the given polar equation: \(r = -4 \cos^2(\theta / 2)\). It represents a function where \(r\) denotes the distance from the origin to a point, and \(\theta\) is the angle from the positive x-axis.

Step 2: Analyze the Equation for Constants and Symmetry

Notice the presence of the negative sign and the square of the cosine function. The negative sign in the equation will cause the plot to be symmetric about the origin (or is equivalent to rotating the graph of \(4\cos^2(\theta/2)\) by \(\pi\) radians).

Step 3: Determine Key Points Using Values of \(\theta\)

Evaluate the equation at key angles to find points for sketching: \(\theta = 0, \pi, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{2\pi}{3}\) etc., to identify changes or symmetry. For example, at \(\theta = 0\), \(r = -4\), confirming a point in the opposite direction of the positive x-axis.

Step 4: Sketch the Symmetric Nature Over a Full Rotation

Begin sketching for a full \(2\pi\) rotation, marking the calculated points repetitively. Since the equation depends on \(\cos^2(\theta/2)\), the values for \(r\) repeat across symmetric intervals like \([0, 2\pi]\) and more.

Step 5: Placing the Key Points on the Polar Grid

On the polar grid, mark the calculated key points for different \(\theta\) values. Plot each point by measuring \(r\) from the origin in the direction given by \(\theta\). Adjust the values considering the negative radius, which inverts the actual position direction.

Step 6: Finish the Curve by Connecting Points

Connect the marked points smoothly to form the complete polar graph. Due to the negative and squared characteristics, the graph will look similar to an inverted, symmetric floral pattern about the pole (origin).

Key Concepts

Graphing Polar Equations

Polar equations describe curves using a distance from the origin (r) and an angle (θ) from the positive x-axis. To graph a polar equation like \( r = -4 \cos^2(\theta/2) \), begin by exploring how \( r \) changes with different values of \( \theta \). This equation implies \

• The distance \( r \) is always negative, causing the plotted point to be in the opposite direction given by \( \theta \).
• The equation involves \( \cos^2(\theta/2) \), showing a dependency on half-angle.
• Max values of \( \cos^2(\theta/2) \) lead to the largest absolute values of \( r \).

To start graphing, choose significant angles such as \( \theta = 0, \pi, \frac{\pi}{2}, \frac{3\pi}{2} \), ensuring to compute \( r \) for each. Then, map these points onto a polar coordinate grid. Make sure to account for the negative sign, as it changes the plotting direction for each point.

Symmetry in Polar Graphs

Symmetry plays a crucial role in simplifying the graphing of polar equations. In the equation \( r = -4 \cos^2(\theta/2) \), symmetry occurs due to both the squaring function and the negative sign.Understanding symmetry helps:

• Identify repeated patterns over each period of \( 2\pi \), which simplifies sketching.
• Predict points without recalculating \( r \) for every angle.

For \( \cos(x) \), symmetry is naturally about both the x-axis and y-axis depending on terms. However, the negative sign here indicates origin symmetry— for each point \((r, \theta)\), its symmetric counterpart also plots at \((-r, \theta + \pi)\).By recognizing these symmetric properties, the plotting process becomes quicker since you can infer additional points and curves from one section of the graph. This understanding helps ensure the entire polar graph maintains its intended look, like a symmetric floral pattern centered at the origin.

Polar Equation Analysis

Analyzing a polar equation seems complex at first but boils down to understanding component effects on \( r \) based on \( \theta \). For \( r = -4 \cos^2(\theta/2) \), the focus lies on:

• Exploring how each trigonometric component influences the graph's shape and position.
• Noting how cosine terms alter amplitude and spacing of points in polar graphs.

Break down the term \( \cos^2(\theta/2) \) as it impacts symmetry and amplitude:

• \( \cos \) reaches its peak value at integers of \( \pi \), leading to expression peaks.
• Squaring the cosine dampens negative values (flattening curve sections).
• The negative sign before the 4 suggests an opposite plot trajectory from positive cosine counterparts.

An effective analysis involves observations such as how variations in \( \theta \) manipulate \( r \), tweaking radius length, positioning, and symmetry across sections of \( 0 \) to \( 2\pi \). By integrating these insights, drawing the curve maps seamlessly practical.