Problem 56 - Precalculus: Functions and Graphs

Question

Find an equation in \(x\) and \(y\) that has the same graph as the polar equation. Use it to help sketch the graph in an \(r \theta\) -plane. $$r\left(r \sin ^{2} \theta-\cos \theta\right)=3$$

Step-by-Step Solution

Answer

The Cartesian equation is \( y^2 = x + 3 \), representing a parabola.

Step 1: Identify Polar Equation

The given polar equation is \( r\left(r \sin ^{2} \theta-\cos \theta\right)=3 \). Our goal is to convert this into Cartesian coordinates (\(x\) and \(y\)).

Step 2: Use Polar to Cartesian Conversions

Recall the conversions: \( x = r \cos \theta \) and \( y = r \sin \theta \). We also know \( r^2 = x^2 + y^2 \) and \( \sin^2 \theta = \frac{y^2}{r^2} \).

Step 3: Substitute Polar Conversions

Substitute the polar conversions into the equation: \[ r\left(r \frac{y^2}{r^2} - \frac{x}{r}\right)=3 \]. This simplifies to \[ (x^2 + y^2)\frac{y^2}{r^2} - x = 3 \].

Step 4: Simplify and Solve for \(r^2\)

From the equation \( (x^2 + y^2)\frac{y^2}{x^2 + y^2} - x = 3 \), simplify to \( y^2 - x = 3 \).

Step 5: Final Cartesian Equation

The equation in Cartesian coordinates is \( y^2 = x + 3 \). This is an equation of a parabola.

Key Concepts

Polar Coordinates

In mathematics, polar coordinates offer a different way to locate points in a plane compared to the traditional Cartesian system. Polar coordinates express a point as a distance from a central point, called the "pole," and an angle from a reference direction. Instead of using two perpendicular coordinates like Cartesian (x, y), polar coordinates use:

• Radius (\( r \)): The distance from the pole (origin) to the point.
• Angle (\( \theta \)): The angle measured from a reference direction, typically the positive x-axis, counterclockwise.

A polar coordinate is represented as (\( r, \theta \)). This form is particularly useful for problems involving circular or radial symmetry. For example, equations or graphs with circles or spirals often simplify in polar form, requiring fewer calculations than using Cartesian coordinates. Polar coordinates change the way we think about positions in a plane, making some problems more intuitive.

Cartesian Coordinates

Cartesian coordinates are based on a grid of horizontal and vertical lines and are represented by the familiar (\( x, y \)) format. This system defines a point in a plane by specifying its distances along two axes, traditionally called the x-axis and y-axis, which intersect at a point called the origin.

• \( x \): The horizontal distance of a point from the origin.
• \( y \): The vertical distance of a point from the origin.

The Cartesian coordinate system is versatile and widely used in various applications from graphing functions to describing geometric shapes. To convert from polar coordinates to Cartesian coordinates, use the formulas:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

By leveraging these relationships, any point described in polar form (\( r, \theta \)) can be transformed into the Cartesian format. This transformation is crucial when analyzing problems that benefit from both representations.

Equation of a Parabola

A parabola is a symmetrical, curved shape that is defined mathematically as the graph of a quadratic function. In Cartesian coordinates, a standard parabola has the general equation \( y^2 = 4px \) or \( x^2 = 4py \), depending on its alignment. The focus-directrix property of a parabola explains its shape: all points are equidistant from a fixed point (the focus) and a line (the directrix).

• The direction of the parabola's opening depends on the signs and positions of the variables in the equation:
• If the \( y^2 \) term is present, the parabola opens horizontally.
• If the \( x^2 \) term is present, it opens vertically.

In the step-by-step solution, the converted equation \( y^2 = x + 3 \) represents a horizontally opening parabola shifted horizontally by 3 units. Understanding how parabolas relate to their equations helps predict their graphical behavior, which is essential when interpreting transformations between Cartesian and polar equations.