Problem 50 - Precalculus with Calculus Previews

Question

The graphs of the equations (a)-(d) represent a rotation of the graph of the given equation. Try sketching these graphs by hand. If you have difficulties, then use a graphing utility. \(r=2+4 \cos \theta\) (a) \(r=2+4 \cos (\theta+\pi / 6)\) (b) \(r=2+4 \cos (\theta-3 \pi / 2)\) (c) \(r=2+4 \cos (\theta+\pi)\) (d) \(r=2+4 \cos (\theta-\pi / 8)\)

Step-by-Step Solution

Answer

Graphs (a) and (c) are counterclockwise rotations by \( \pi/6 \) and \( \pi \) radians, respectively, while (b) and (d) are clockwise by \( 3\pi/2 \) and \( \pi/8 \) radians.

Step 1: Understanding the Original Equation

The original equation is given as \( r = 2 + 4 \cos \theta \), which represents a polar graph known as a limaçon. The graph's specific shape is affected by the coefficients and constants within the equation, typically creating a curve with an inner loop, dimple, or without loops.

Step 2: Recognizing the Effect of Angle Shifts

The angle shift \( \theta + \alpha \) or \( \theta - \alpha \) translates to a rotational transformation of the original graph. Positive shifts represent counterclockwise rotations, while negative shifts indicate clockwise rotations.

Step 3: Graph (a) Analysis

For graph \( r = 2 + 4 \cos (\theta + \pi / 6) \), the graph will rotate counterclockwise by \( \frac{\pi}{6} \) radians. Thus, this graph is a rotational transformation of the original limaçon by 30 degrees counterclockwise.

Step 4: Graph (b) Analysis

The equation \( r = 2 + 4 \cos (\theta - 3\pi/2) \) suggests a clockwise rotation by \( \frac{3\pi}{2} \) radians (or 270 degrees). This shifts the easy-to-recognize features of the graph such as the dimple position accordingly.

Step 5: Graph (c) Analysis

The transformation is expressed by \( r = 2 + 4 \cos (\theta + \pi) \). This indicates a half-turn, or \( \pi \) radians (180 degrees) counterclockwise, effectively flipping the graph around the origin.

Step 6: Graph (d) Analysis

Similarly, the equation \( r = 2 + 4 \cos (\theta - \pi/8) \) rotates the graph clockwise by \( \frac{\pi}{8} \) radians, or 22.5 degrees. The graph's features pivot slightly from their original alignment.

Key Concepts

Polar Graph

Polar coordinates are a unique way of graphing that uses a radius, \( r \), and an angle, \( \theta \), as its key components instead of the traditional Cartesian \( x \) and \( y \) coordinates. This method is particularly useful for graphing circular and spiraling patterns.

In a polar graph, each point is determined by how far it is from the origin (the pole) and the angle it makes with a fixed direction, typically the positive x-axis. This system allows for the elegant plotting of curves like circles, spirals, and more complex figures such as limaçons.

• A positive \( r \) value indicates movement away from the origin, following the angle \( \theta \).
• A polar graph requires mapping the angle first, then extending out to reach the radius.

When sketching polar graphs, understanding the relationship between \( r \) and \( \theta \) helps to plot points and trace the curve effectively.

Limaçon

A limaçon is a unique type of polar graph characterized by its distinct shapes that may include loops, dimples, or a regular oval shape. It’s interestingly named after the French word for "snail" due to its resemblance. The equation given in the original exercise, \( r = 2 + 4 \cos \theta \), is an example of a limaçon.

Depending on the parameters in its equation, a limaçon can take different forms:

• **With a loop**: Happens when the coefficient of \( \cos \theta \) is greater than the constant term.
• **Cardioid-shaped**: Occurs when both terms are equal.
• **Dimpled or convex**: Arises when the constant term exceeds the \( \cos \theta \) coefficient.

A limaçon is graphical representation offers opportunities to witness intriguing movements and shapes, which expand into more dynamic figures when angle shifts are introduced.

Angle Shift

Angle shifts in polar equations like \( r = a + b \cos(\theta + \alpha) \) modify the angle \( \theta \) by adding or subtracting a certain angle \( \alpha \). This transformation results in rotating the original graph around the origin without altering its size or shape. Consider each angle shift as a simple adjustment of orientation.

Depending on the sign of the angle shift, the graph rotates in distinct directions:

• **Positive shifts** like \( \theta + \pi/6 \) result in counterclockwise rotations—think of this as turning left.
• **Negative shifts** such as \( \theta - 3\pi/2 \) cause clockwise rotations—turning right in the visual representation.

These rotational transformations help in visualizing the beauty of polar graphs by showing how the orientation of features like loops or dimples change depending on the angle shift applied.

Rotational Transformation

Rotational transformations are pivotal in polar graphs, as they dictate the new orientation of the graph's design without manipulating its size. Picture the entire graph spinning around the origin either clockwise or counterclockwise as per the specified angle shift.

Examining the given transformations in the exercise, we witness significant orientation changes:

• A **half-turn rotation** like \( \theta + \pi \) , flips the graph upside down.
• A **lesser angle** such as \( \theta - \pi/8 \) imparts a slight clockwise tilt.
• A **right angle rotation**,\( \theta - 3\pi/2 \), results in a broader turn, aligning the graph differently with the axis.

These transformations add dynamism to polar graphs making them versatile and illustrative, each rotation offering a new perspective on the graph's symmetrical beauty.