Question

Use a graphing utility to graph the parabolas \(r=\frac{d}{1+\cos \theta}\) for \(d=0.25,0.5,1,2,3,\) and 4 on the same set of axes. Explain how the shapes of the curves vary as \(d\) changes.

Short answer

In summary, the polar equation \(r=\frac{d}{1+\cos \theta}\) generates curves with a loop on one side of the polar axis. The size of the loop and the distance from the origin increase as the value of \(d\) increases, while the basic shape and direction of the loop remain constant. This variation in the shape of the curves is primarily due to the increasing numerator (d) of the polar equation, which results in a larger value of \(r\) for a fixed angle \(\theta\).

Step-by-step solution

Write the polar equation for different values of d

First, write the polar equation \(r=\frac{d}{1+\cos \theta}\) for \(d=0.25, 0.5, 1, 2, 3,\) and 4. 1. When \(d=0.25\), we have \(r=\frac{0.25}{1+\cos \theta}\). 2. When \(d=0.5\), we have \(r=\frac{0.5}{1+\cos \theta}\). 3. When \(d=1\), we have \(r=\frac{1}{1+\cos \theta}\). 4. When \(d=2\), we have \(r=\frac{2}{1+\cos \theta}\). 5. When \(d=3\), we have \(r=\frac{3}{1+\cos \theta}\). 6. When \(d=4\), we have \(r=\frac{4}{1+\cos \theta}\).

Plot the curves using a graphing utility

Now, we will use a graphing utility to plot the curves for the given polar equations. You can use any graphing utility like Desmos, Geogebra, or a graphing calculator. After plotting all the curves with different values of \(d\), we have the desired graphs.

Observe the shapes and their variations

By looking at the graphs plotted, we can observe the following variations: 1. As \(d\) increases, the size of the curve (the distance from the origin) increases. 2. The shape of the curve is similar for all the values of \(d\). Each curve has a loop on one side of the polar axis. 3. The direction of the loop does not change with \(d\). For every curve, the loop is on the left side of the polar axis.

Explanation of the variation in shapes

The variation in the shapes of the curves is mainly due to the values of \(d\). As \(d\) increases, the numerator of the polar equation increases, leading to an increase in the value of \(r\) for a fixed angle \(\theta\). Since \(r\) represents the distance of the curve from the origin, the loop becomes larger as \(d\) increases and the curve moves further away from the origin. The direction and the basic shape of the loop remain the same because the denominator part of the equation, \(1+\cos \theta\), does not change with \(d\). This is the reason behind the variation in the shape of curves as \(d\) changes.

Key concepts

Graphing Utility

Graphing utilities are powerful tools that help visualize mathematical equations. When working with a polar equation, using a graphing utility like Desmos or Geogebra simplifies the process of plotting and interpreting graphs.
These utilities allow you to input equations and immediately see their visual representation, which is especially useful for polar equations where the relationship between the radius and angle needs clear depiction.
For our exercise, we input the equation \(r=\frac{d}{1+\cos \theta}\) for various values of \(d\). Each graph provides insight into how changes in the parameter affect the curve's size and position. By observing multiple graphs simultaneously, it becomes easier to draw conclusions about the relationships between variables.
Using a graphing utility is like opening a window into the mathematical world, making complex relationships more tangible and easier to understand.

Polar Equation

Polar equations express the relationship between the radius \(r\) and the angle \(\theta\) on a coordinate plane. These equations are typically used in scenarios where circular or angular relationships are more natural than rectangular coordinates.
The equation \(r=\frac{d}{1+\cos \theta}\) represents a type of conic section -- specifically, a limaçon -- when expressed in polar coordinates. Unlike Cartesian equations with x and y values, polar equations use \(r\) to describe the radial distance from the origin and \(\theta\) to represent the angular orientation.
Understanding how polar equations work involves recognizing how changes in parameters like \(d\) affect the size and shape of the graph. As \(d\) increases, \(r\) grows for every \(\theta\), expanding the shape outward, which ultimately provides visual insight into the mathematical relationship described by the equation.

Conic Sections

Conic sections are the curves obtained by intersecting a cone with a plane. This produces various shapes such as circles, ellipses, parabolas, and hyperbolas, each with unique properties and equations.
In polar coordinates, a similar concept involves the limaçon, which is derived from the polar equation \(r=\frac{d}{1+\cos \theta}\). This reveals characteristics akin to conic sections, where the shape and size depend on the parameter \(d\).
The variation in shapes as \(d\) changes highlights the dynamic nature of conic sections. For lower values of \(d\), the loops are smaller, but as \(d\) increases, the loops enlarge and the curve stretches outward. Despite these changes, the curves maintain a critical symmetry around the polar axis, reflecting the fundamental nature of conic sections.

• Conics in polar coordinates present a vivid depiction of mathematical beauty and symmetry.
• Alterations in parameters like \(d\) illustrate the flexibility and diverse possibilities within these geometric figures.