Question

Use a graphing utility to graph \(r=\frac{1}{1-\cos \theta}\).

Short answer

The graph is a parabola with its focus at the pole, opening towards the right.

Step-by-step solution

Understand the Equation

The given equation is in a polar form where \(r\) is defined as \(r = \frac{1}{1 - \cos \theta}\). This equation represents a conic section with a focus at the pole.

Identify the Type of Graph

The equation \(r = \frac{1}{1 - \cos \theta}\) is in the form \(r = \frac{ed}{1 - e\cos \theta}\) where \(e = 1\). This indicates that the graph is a parabola with its focus at the pole and the directrix perpendicular to the polar axis.

Set Up the Graphing Utility

Open any graphing utility capable of plotting polar equations, such as Desmos or a graphing calculator. Input the function \(r = \frac{1}{1 - \cos \theta}\) into the utility.

Graph the Equation

Plot the function and observe the graph on the polar coordinate system. The graph should resemble a parabola oriented along the polar axis with a vertex at \(\theta = 0\) and opening rightward.

Analyze the Graph

Check the symmetry and shape of the graph. For \(r = \frac{1}{1 - \cos \theta}\), it should be symmetrical about the polar axis (the horizontal axis in polar coordinates).

Key concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These curves include circles, ellipses, parabolas, and hyperbolas. Each type of conic section has a unique equation that describes its shape and position:

• A circle is the set of all points equidistant from a center point.
• An ellipse is similar to a circle but stretched along two axes.
• A parabola includes points equidistant from a fixed point (focus) and a line (directrix).
• A hyperbola consists of two separate curves or branches.

Conic sections are particularly interesting in polar coordinates because they can exhibit unique properties. For instance, the parabola, as seen in this exercise, has its focus at the origin, showing how naturally it fits into the polar coordinate framework. Understanding how to represent these sections in polar form is crucial for applications in fields such as astronomy, physics, and engineering.

Graphing Utility

A graphing utility is a tool that allows users to visualize mathematical equations and functions in various coordinate systems, including Cartesian and polar. These utilities make it easier to analyze and interpret complex mathematical concepts and solve equations visually. For graphing polar equations like the one in this exercise, you can use tools such as:

• Desmos, an online graphing calculator that is user-friendly and highly interactive.
• Graphing calculators, such as those made by Texas Instruments, which provide options for visualizing functions in polar coordinates.

These utilities allow you to input the polar equation quickly, adjust the viewing window, and observe the graph as it updates in real-time. This immediate visual feedback is immensely valuable in understanding the property of the graph, for example, observing how the parabola opens and its symmetry.

Parabola in Polar Form

A parabola in polar form is a representation of the curve with respect to a point called the focus, situated at the pole, and a directrix, which is a line perpendicular to the polar axis. The given equation, \( r = \frac{1}{1 - \cos \theta} \), reflects a parabola with its focus at the pole of the polar coordinate system. Key features of this polar parabola include:

• The focus is at the origin, making it distinct in appearance compared to its Cartesian counterpart.
• The directrix is a line perpendicular to the polar axis, influencing the shape and direction in which the parabola opens.
• The symmetry axis, in this case, is the polar axis, indicating that the graph is symmetrical about this line.

Understanding these features helps in recognizing the parabola's opening and the distribution of points around the focus, which is crucial in applications like satellite dish design and optical instruments, where paraboloids collect or reflect light.