Problem 48 - Precalculus Mathematics for Calculus

Question

Graph the family of polar equations \(r=1+c \sin 2 \theta\) for \(c=0.3,0.6,1,1.5,\) and \(2 .\) How does the graph change as \(c\) increases?

Step-by-Step Solution

Answer

The petals become more pronounced and larger as \( c \) increases.

Step 1: Understanding the Polar Equation

The given polar equation is \( r = 1 + c \sin(2\theta) \). This equation forms a type of graph called a rose curve. The behavior of the graph depends on the parameter \( c \), and the coefficient of \( \theta \) inside the sine function, which is 2 in this case, determines the number of petals.

Step 2: Determine the Number of Petals

Since the equation is in the form \( r = 1 + c \sin(n\theta) \) with \( n = 2 \), the number of petals is double the value of \( n \) when \( n \) is even. Therefore, for each value of \( c \), the graph will have 4 petals.

Step 3: Graph the Equation for Various Values of \( c \)

For each specified value of \( c \) (0.3, 0.6, 1, 1.5, 2), the rose curve is initially a circle when \( c = 0 \) and begins to form lobes (petals) as \( c \) increases. For example:- When \( c = 0.3 \), the curve slightly deviates into a rose shape.- When \( c = 0.6 \), the petals become more pronounced.- When \( c = 1 \), the petals are fully visible, producing a symmetrical rose with 4 petals.- When \( c = 1.5 \), the size of the petals increases.- When \( c = 2 \), the petals expand further, showing the extremes of the curve's shape.Each graph becomes more distinctly defined as \( c \) increases.

Step 4: Analyze Changes in the Graph

As \( c \) increases from 0.3 to 2, the amplitude of the sine wave component (\( c \sin(2\theta) \)) increases, which results in petals that extend further from the center. The overall shape becomes more exaggerated with larger values of \( c \), indicating that the magnitude of the oscillation increases, causing more pronounced lobes.

Key Concepts

Rose Curve

A rose curve is a fascinating shape formed by plotting a specific type of polar equation. The name comes from its characteristic petal shapes, much like the petals of a rose. In polar coordinates, a rose curve typically follows the form \( r = a + b \sin(n\theta) \) or \( r = a + b \cos(n\theta) \). In our case, the equation \( r = 1 + c \sin(2\theta) \) produces a rose curve.The number of petals in a rose curve is determined by the coefficient \( n \) in \( n\theta \). If \( n \) is even, like in our exercise where \( n = 2 \), the number of petals is \( 2n \). Thus, you will see four petals for any value of \( c \) since \( 2 \times 2 = 4 \). If \( n \) were odd, the number of petals would equal \( n \). Understanding this pattern helps predict the rose's structure without plotting it fully.

Graphing Polar Coordinates

Graphing polar coordinates involves plotting points based on their distance from the origin and the angle formed with the positive x-axis. This differs from Cartesian graphs where points are plotted based on x and y coordinates.In polar graphs, each point is represented as \( (r, \theta) \). The radius \( r \) is the distance from the parabola's origin, while \( \theta \) is the angle. For a given equation like \( r = 1 + c \sin(2\theta) \), you calculate the radius at various angles to observe the graph's full shape.As you increase \( \theta \), the polar equation provides different radii \( r \), creating the graph's notable lobes (or petals). This approach allows for smooth curves and unique shapes, offering a different perspective from traditional graphing methods.

Effect of Parameters on Graph Shape

The parameter \( c \) in the equation \( r = 1 + c \sin(2\theta) \) significantly impacts the graph's appearance. Essentially, it modifies the amplitude of the sine function, which affects the extent to which the petals stretch from the origin.

• When \( c = 0.3 \), the difference is minimal, and the graph keeps mostly circular with slight petal formation.
• As \( c \) increases to 0.6, the petals become a little more pronounced, indicating an increased amplitude.
• Further increasing \( c \) to 1 results in clearly visible petals, forming a symmetrical rose shape.
• Values like \( c = 1.5 \) and \( c = 2 \) amplify the petals even more, pushing them outwards and creating more striking features.

This demonstrates a key property: larger values of \( c \) lead to more dramatic variations and a bolder curve as the petals become more distinct. Understanding parameter impact is crucial for mastering polar graph behavior and predicting curve shapes.