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Chapter 1: Problem 162

Convert the rectangular equation to polar form and sketch its graph. $$ y^{2}=4 x $$

Short Answer

Expert verified
The polar form is \( r\sin^2(\theta) = 4\cos(\theta) \); it sketches as a rightward-opening parabola.

Step by step solution

01

Understand the Given Equation

The given equation is a parabola, represented in rectangular coordinates as \( y^2 = 4x \). In this form, the equation describes a parabola opening to the right, with its vertex at the origin (0,0).
02

Recall Polar Coordinates

In polar coordinates, each point is described by its distance from the origin, \( r \), and the angle \( \theta \) it makes with the positive x-axis. The relationships between rectangular and polar coordinates are \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).
03

Substitute Rectangular to Polar

Substitute \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \) into the equation \( y^2 = 4x \). This results in \( (r\sin(\theta))^2 = 4(r\cos(\theta)) \).
04

Simplify the Polar Equation

Expand and simplify: \( r^2\sin^2(\theta) = 4r\cos(\theta) \). To simplify, divide both sides by \( r \) (assuming \( r eq 0 \)) to get \( r\sin^2(\theta) = 4\cos(\theta) \). This is the polar form of the equation.
05

Analyze and Sketch the Graph

The polar equation \( r\sin^2(\theta) = 4\cos(\theta) \) describes a conic section. Since it originates from \( y^2 = 4x \), it remains a parabola but expressed in terms of polar coordinates. The sketch will be identical to the parabola \( y^2 = 4x \) opening in the positive direction along a radial line.

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Key Concepts

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

Rectangular Coordinates
Rectangular coordinates are a way to represent points on a plane using two numbers: the x-coordinate and the y-coordinate. These numbers tell how far along the axis a point is from the origin, which is the center point (0,0) on the Cartesian plane. This system is also known as Cartesian coordinates.

In rectangular coordinates:
  • The x-coordinate shows how far left or right a point is from the x-axis.
  • The y-coordinate shows how far up or down a point is from the y-axis.
Imagine a grid on a piece of paper with horizontal and vertical lines. Each line represents a unit distance. When you place a point on this grid, you can see exactly where it falls relative to the origin by counting the number of lines it moves horizontally and vertically. Rectangular coordinates are incredibly useful for plotting and analyzing points and equations like lines, circles, and in our case, parabolas.
Parabola
A parabola is a specific type of curve on a graph, shaped like an arch. It can open up, down, left, or right depending on how its equation is structured. In our exercise, the equation given is \( y^2 = 4x \), which represents a parabola.

Key characteristics of a parabola:
  • It has a single peak or vertex, which in the equation \( y^2 = 4x \) is at the origin (0,0).
  • It is symmetric, meaning one side is a mirror of the other.
  • This particular parabola opens to the right, shown by the equation having \( x \) alone on one side.
Parabolas are widely found in the real world, from the paths of thrown objects to the designs of satellite dishes. In math, understanding their shape and equation helps us visualize and solve problems involving energy, motion, and light.
Conic Sections
Conic sections are shapes created as a plane intersects a cone. These sections include circles, ellipses, parabolas, and hyperbolas. Each conic section has specific equations and forms based on this intersection.

What makes parabolas a conic section:
  • A parabola occurs when a plane is parallel to the slant of the cone.
  • These can be expressed in standard form, such as \( y^2 = 4x \) for a parabola, showing their unique structure.
  • They are important in fields like physics, engineering, and astronomy.
In our case, the equation \( y^2 = 4x \) is a classic example of a parabola as a conic section. Moving from rectangular to polar coordinates involves translating these forms, integrating different mathematical perspectives to view these sections in multiple dimensions.

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Most popular questions from this chapter

For the parametric curve whose equation is \(x=4 \cos \theta, y=4 \sin \theta, \quad\) find the slope and concavity of the curve at \(\theta=\frac{\pi}{4}\).

The trajectory of a bullet is given by \(x=v_{0}(\cos \alpha) t y=v_{0}(\sin \alpha) t-\frac{1}{2} g t^{2}\) where \(v_{0}=500 \mathrm{~m} / \mathrm{s}, \quad g=9.8=9.8 \mathrm{~m} / \mathrm{s}^{2}, \quad\) and \(\alpha=30\) degrees. When will the bullet hit the ground? How far from the gun will the bullet hit the ground?

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{3}, \quad y=3 \ln t, t \geq 1 $$

Find \(d y / d x\) at the value of the parameter. $$ x=\cos t, \quad y=\sin t, \quad t=\frac{3 \pi}{4} $$

Each set of parametric equations represents a line. Without eliminating the parameter, find the slope of each line. $$ x=-5 t+7, \quad y=3 t-1 $$
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