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Chapter 6: Problem 31

Graphing a Polar Equation, use a graphing utility to graph the polar equation. Identify the graph. $$r=\frac{4}{3-\cos \theta}$$

Short Answer

Expert verified
The polar plot for the given equation \(r=\frac{4}{3-\cos \theta}\) is a Limacon, a dimple-shaped graph.

Step by step solution

01

Identify the polar equation

The given polar equation is \(r=\frac{4}{3-\cos \theta}\)
02

Graphing the polar equation

To graph the polar equation, input the equation into the graphing utility. For each value of theta, the graphing utility will compute the corresponding `r` value by substituting theta into the equation, and represent this as a point in polar coordinates. The range of theta is conventionally from 0 to \(2\pi\) or 0 to 360 degrees.
03

Interpreting the polar plot

Once the polar plot is sketched, note whether the plot is symmetric about the x-axis, the y-axis, or the origin. This knowledge could help to identify the graph. For the given equation, the plot will be a Limacon, a dimple-shaped graph, since it is of the form \(r=a + b cos \theta \)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a handy digital tool used to visualize mathematical equations. It's particularly useful for students and educators to plot graphs and better understand mathematical behavior. In the case of polar equations, graphing utilities simplify complex calculations and display them visually.

Here鈥檚 how it works:
  • First, you input the equation into the graphing utility.
  • As the utility processes the equation, it calculates the values of variables for numerous points. For polar coordinates, it translates `\(\theta\)` into a radius `\(r\)`.
  • It then visualizes these calculations on a virtual graph.
This visual representation makes it much easier to understand intricate shapes and patterns, like those found in polar graphs. It provides instant feedback, which is crucial for learning and exploring mathematical concepts more deeply.
Polar Coordinates
Polar coordinates offer a unique way of expressing points in a plane. Unlike the Cartesian system, which uses `\(x\)` and `\(y\)` to locate a point, polar coordinates rely on two elements: `\(r\)` and `\(\theta\)`.

Here's a simple breakdown:
  • The distance `\(r\)`, which is the radial coordinate, measures how far the point is from the origin.
  • The angle `\(\theta\)` denotes the angular coordinate, showing the direction of the point relative to the positive x-axis.
These coordinates are particularly useful for equations that describe circular or spiral patterns, making polar coordinates particularly useful for showcasing complex curved structures. By understanding these, students can easily navigate and comprehend the various forms that polar graphs can take.
Limacon Graph
The Limacon graph is a fascinating type of polar graph. It is derived from equations of the form `\(r = a + b \cos \theta\)` or `\(r = a + b \sin \theta\)`. These graphs have a variety of forms depending on values assigned to `\(a\)` and `\(b\)`.

Here鈥檚 what makes Limacon graphs unique:
  • They can appear with a dimple, a loop, or even take a cardioid shape.
  • Symmetry often plays a key role, maybe about the x-axis, y-axis, or both.
  • The distinctive shape arises from the interplay between the constants `\(a\)` and `\(b\)`.
The graph of `\(r=\frac{4}{3-\cos \theta}\)` represents a Limacon with a dimple. Mastery of identifying these graphs and recognizing their traits helps students understand the diverse and rich nature of polar equations. It鈥檚 a visual adventure in mathematics, inspiring curiosity and deeper learning.

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Most popular questions from this chapter

Verifying a Polar Equation Show that the polar equation of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \quad\) is \(\quad r^{2}=\frac{b^{2}}{1-e^{2} \cos ^{2} \theta}\)

A circle and a parabola can have \(0,1,2,3,\) or 4 points of intersection. Sketch the circle \(x^{2}+y^{2}=4 .\) Discuss how this circle could intersect a parabola with an equation of the form \(y=x^{2}+C .\) Then find the values of \(C\) for each of the five cases described below. Use a graphing utility to verify your results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$r=-3 \sin \theta$$

Determine whether the statement is true or false. Justify your answer. If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

An archer releases an arrow from a bow at a point 5 feet above the ground. The arrow leaves the bow at an angle of \(15^{\circ}\) with the horizontal and at an initial speed of 225 feet per second. (a) Write a set of parametric equations that model the path of the arrow. (See Exercises 93 and 94 .) (b) Assuming the ground is level, find the distance the arrow travels before it hits the ground. (Ignore air resistance.) (c) Use a graphing utility to graph the path of the arrow and approximate its maximum height. (d) Find the total time the arrow is in the air.
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