Problem 7 - The Calculus with Analytic Geometry

Question

Draw a sketch of the graph of the given equation.\(r=5\)

Step-by-Step Solution

Answer

The graph is a circle with radius 5 centered at the origin.

Step 1: Identify the Equation Type

Recognize that the equation given, \(r = 5\), is a polar equation. In this form, \(r\) represents the radius from the origin to a point on the graph.

Step 2: Interpret the Equation

Understand that \(r = 5\) means all points are at a distance of 5 units from the origin, irrespective of the angle \(\theta\).

Step 3: Draw the Graph

Since all points are at a constant distance of 5 units from the origin, the graph will be a circle with a radius of 5 centered at the origin. Use a compass to draw a circle with a radius of 5 units.

Step 4: Label the Graph

Mark the center of the circle at the origin (0,0) and label the radius as 5 units.

Key Concepts

Polar Equations

A polar equation is a mathematical expression that defines a relationship between the radius \(r\) and the angle \(\theta\) in the polar coordinate system. The polar coordinate system is different from the Cartesian (rectangular) coordinate system.

This system specifies each point uniquely by two numbers:

• Radius \(r\), the distance from the origin (pole)
• Angle \(\theta\), the angle from the positive x-axis (usually in radians or degrees)

In your exercise, the given equation is \(r = 5\). Here, \(r\) indicates the constant radius of 5 units from the origin. The polar equation helps you understand the shape and position of the graph. In this case, \(r = 5\) represents points that are always 5 units away from the origin irrespective of the angle \(\theta\).

Graphing Techniques

To graph a polar equation, follow some key techniques to ensure accuracy:

• Identify the equation type: Recognize whether it’s a standard form, spiral, or another type.
• Interpret the equation: Determine what it signifies about the graph’s shape.
• Use appropriate graphing tools: For precise drawing, a compass is ideal for circles.
• Mark significant points and angles: This aids in correctly positioning the graph.

In the exercise, once you identify that \(r = 5\) represents a set of points 5 units from the origin, you can confidently proceed to draw a circle. Use a compass centered on the origin (0,0) adjusting to the 5 units radius. This forms a perfect circle centered at the origin.

Circular Graphs

Circular graphs are a common type of graph in polar coordinates. They represent all points that are an equal distance from a single point, the origin.

In polar coordinates, a circle can be defined by an equation of the form \(r = a\), where \(a\) is the radius of the circle. For example, \(r = 5\) means the radius is 5 units.
This implies that no matter the angle \(\theta\), the distance \(r\) from the origin remains 5 units.
Circular graphs have distinct features:

• Symmetry: They are perfectly symmetrical about the origin.
• Radius: Constant radius from the center to all points on the circle.
• Simplicity: They are simple to graph and interpret once understood.

Understanding circular graphs aids in recognizing and sketching them quickly when presented with such polar equations.