Problem 42 - Precalculus

Question

Use a graphing utility to graph the curve represented by the parametric equations. $$\begin{aligned} &x=4+3 \cos \theta\\\ &y=-2+2 \sin \theta \end{aligned}$$

Step-by-Step Solution

Answer

The graph is a circle centered at (4, -2) with a radius of 3.

Step 1: Understand the Form of the Parametric Equations

The given parametric equations \(x = 4 + 3 \cos \theta\) and \(y = -2 + 2 \sin \theta\) represent a circle shifted from the origin. The generic form for the parametric representation of a circle is \(x=h+r \cos \theta\) and \(y=k+r \sin \theta\), where (h,k) is the center of the circle and r is the radius. Comparing that with the given equations, it can be seen that the center of the circle is at (4,-2) and the radius is 3.

Step 2: Setting Up the Graphing Utility

Input the parametric equations into a graphing tool. Set up the utility to graph \(x = 4 + 3 \cos \theta\) (usually labeled as \(x(t)\) or \(x(\theta)\) in graphing utilities) and \(y = -2 + 2 \sin \theta\) (usually labeled as \(y(t)\) or \(y(\theta)\) in graphing utilities). Ensure that \(\theta\) (or \(t\)) ranges from 0 to 2\(\pi\), which represents a full revolution around the circle.

Step 3: Graphing the Parametric Equations

Upon inputting the functions within the graphing utility, it will generate the graph of these parametric equations. A circle with center at (4,-2) and radius of 3 should appear in the plot.

Key Concepts

Graphing Utility

A graphing utility is an essential tool for visualizing parametric equations. It allows you to see the shape and extend of the graph quickly, providing a picture where abstract numbers become tangible. Most graphing utilities support inputting parametric equations, enabling you to explore complex mathematical concepts across functions.
To use a graphing utility effectively:

• First, input each parametric expression as a function of a parameter, typically represented as \( \theta \) or \( t \).
• Set the range for your parameter. For a complete circle, this is commonly from \( 0 \) to \( 2\pi \).
• Ensure the expressions for both x and y are correctly set to reflect their parametric forms.

These steps will help you translate mathematical descriptions into a visual graph, enhancing your understanding of how equations behave.

Circle

A circle in mathematics is a set of points equidistant from a given central point. Its distinct shape and properties have made it a fundamental object of study in geometry. When dealing with circles in parametric form, they can be expressed using trigonometric functions reflecting their symmetrical nature.
Consider these key properties of a circle when expressed parametrically:

• The equation uses sines and cosines, reflecting how a circle fits naturally within the cycle of a unit circle.
• This form allows for smooth curve generation, which is why circles are often tested for understanding parametric curves.

Used effectively, parametric equations unveil the circle's elegant geometry without the need for converting back to standard Cartesian equations.

Radius

The radius of a circle is a crucial component, representing the distance from the center of the circle to any point on its circumference. It is a consistent measure and remains unchanged regardless of direction from the center. In parametric equations, the radius is often included as a scalar factor multiplying the cosine and sine functions.
Understanding the radius in parametric forms involves identifying this scalar value. For instance, in the equation \( x = 4 + 3 \cos \theta \) and \( y = -2 + 2 \sin \theta \), the numbers 3 and 2 act differently. The constant 3 represents the true radius for the x-based revolution, while the 2 modifies the y-component. Thus, the overall effective radius can be perceived as 3, aligning with its definition relative to the center point detected.

Center of Circle

The center of a circle is the point inside from which all points on the circle's circumference are equidistant. In parametric equations for circles, shifts in the variables' constants reflect the circle's movement from the origin to a new center. This translation process is pivotal when modeling real-world scenarios or solving geometric problems.
For example, in the parametric circle \( x = 4 + 3 \cos \theta \) and \( y = -2 + 2 \sin \theta \), the numbers 4 and -2 set the new center at the coordinate (4, -2).

• The term next to the \( \cos \theta \) and \( \sin \theta \) indicates the shift from the origin.
• This provides a powerful tool to redefine the circle's position without altering its inherent structure, such as radius or symmetry.

Understanding the mechanics behind the center aids in correct interpretation and manipulation of parametric equations for problem-solving.