Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 27-44: 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

Equation in rectangular form: \( y^2 - x = 3 \), a rightward opening parabola.

Step 1: Convert the Polar Equation to Rectangular Form

Given the polar equation \( r(r \sin^2 \theta - \cos \theta) = 3 \), we will first use the known polar-to-rectangular coordinate conversions.Recall that \( r^2 = x^2 + y^2 \), \( \sin \theta = \frac{y}{r} \), and \( \cos \theta = \frac{x}{r} \).Substitute \( r^2 \sin^2 \theta = (r \sin \theta)^2 = y^2 \) and \( r \cos \theta = x \) into the equation.So, \( r \sin^2 \theta = \frac{y^2}{r} \) and \( \cos \theta = \frac{x}{r} \). The equation becomes:\( r \left( \frac{y^2}{r} - \frac{x}{r} \right) = 3 \).Simplify it to find: \( y^2 - x = 3 \).

Step 2: Graph the Resulting Equation

The equation in rectangular form is \( y^2 - x = 3 \), which simplifies to \( y^2 = x + 3 \).This is a standard equation for a parabola that opens to the right. In the \(xy\)-plane, the vertex of this parabola is at the point \((-3, 0)\).The parabola opens to the right due to the positive \(x\) term on the right side of the equation.Plot a few points such as \((0, \sqrt{3})\), \((0, -\sqrt{3})\), and \((1, 2)\) to sketch the parabola accurately on the graph.

Key Concepts

Polar Coordinates

Polar coordinates are a way of representing a point in a plane using a distance and an angle. Unlike rectangular coordinates, which use a horizontal and vertical distance from the origin (x and y), polar coordinates express a point as

• a radius or distance from the origin, denoted as \( r \)
• an angle \( \theta \) measured from the positive x-axis

This system is particularly useful in situations where relationships are more easily expressed in terms of angles and distances, such as in circular or spiral patterns.
When converting from polar to rectangular coordinates, it's important to remember these transformations:

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

These expressions help translate a point from polar coordinates into the more familiar x and y coordinates. In our exercise, these transformations were used to convert the equation from polar form to rectangular form, enabling us to further analyze the curve in the Cartesian plane.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, describe a point in a plane with two numbers:

• x, the horizontal distance from the origin
• y, the vertical distance from the origin

This system is widely used in many areas of mathematics and physics due to its straightforward representation of shapes and locations in a plane.
In the process of solving the given equation, converting polar coordinates to rectangular coordinates allowed us to recognize the familiar form of a parabola. The expression \( r (r \sin^2 \theta - \cos \theta) = 3 \) was converted into \( y^2 - x = 3 \), which is a standard quadratic equation that can be easily graphed and understood in the x-y plane.
Understanding how to move between these systems can simplify problem-solving processes, especially when dealing with graphs or curves that are straightforward in one system over the other.

Parabola Graphing

Graphing a parabola using its equation is crucial for visualizing its shape and key features. In the rectangular form, the given equation \( y^2 = x + 3 \) describes a parabola with several important characteristics.
The vertex of the parabola is located at the point

• \((-3, 0)\), signifying the lowest or highest point of the parabola for parabolas that open right or left

This parabola opens to the right, as indicated by the positive sign of the \(x\) term.
Important points such as

• \((0, \pm\sqrt{3})\)
• \((1, 2)\)

can be plotted to gain an accurate sketch of the curve.
Identifying whether a parabola opens upwards, downwards, right, or left is key to plotting a clear and correct graph. These points define its width and help in sketching an accurate representation in the Cartesian plane, aiding in further analysis or practical applications.