Problem 109 - A Graphical Approach to Precalculus with Limits

Question

Describe the graph of the equation as either a circle or a parabola with horizontal axis of symmetry. Then determine two functions, designated by \(y_{1}\) and \(y_{2},\) such that their union will give the graph of the given equation. Finally, graph \(y_{1}\) and \(y_{2}\) in the same viewing rectangle. $$x^{2}+y^{2}=100$$

Step-by-Step Solution

Answer

The graph is a circle. Functions: \(y_1 = \sqrt{100 - x^2}\) and \(y_2 = -\sqrt{100 - x^2}\).

Step 1: Identify the Type of Equation

The given equation is \(x^2 + y^2 = 100\). This equation has squared terms for both \(x\) and \(y\) and is in the form of \(x^2 + y^2 = r^2\). This equation represents a circle centered at the origin \((0,0)\) with a radius of \(r = 10\), since \(10^2 = 100\).

Step 2: Describe Graph and Axis

A circle doesn't have an axis of symmetry like a parabola, which typically has a horizontal or vertical axis of symmetry. Hence, the given graph is a circle and not a parabola.

Step 3: Express y in Terms of x

Re-arrange the equation \(x^2 + y^2 = 100\) to express \(y\):\(y^2 = 100 - x^2\).Next, solve for \(y\):\(y = \pm \sqrt{100 - x^2}\), resulting in two functions: \(y_{1} = \sqrt{100 - x^2}\) and \(y_{2} = -\sqrt{100 - x^2}\).

Step 4: Plot y1 and y2 on a Graph

Plot \(y_1 = \sqrt{100 - x^2}\) and \(y_2 = -\sqrt{100 - x^2}\) on the same set of axes. \(y_1\) will represent the upper semicircle, while \(y_2\) will represent the lower semicircle. Together, they form the complete circle of radius 10 centered at the origin.

Key Concepts

Equation of a Circle

The equation of a circle can be easily identified by its format, which is typically given as \(x^2 + y^2 = r^2\). This is the standard form representing a circle centered at the origin, with \(r\) being the radius.
Understanding this equation is crucial. Here, both \(x\) and \(y\) terms are squared, without linear terms.
This symmetry reflects the balanced nature of a circle, which is uniformly shaped from its center point.

• The "\(x^2 + y^2\)" indicates no shifts along the axes, meaning the circle is perfectly centered at \((0,0)\).
• The "\(r^2\)" is the square of the radius, giving the size of the circle. In the provided equation \(x^2 + y^2 = 100\), the value \(r^2 = 100\), which gives \(r = 10\).

This radius defines the distance from any point on the circle to the center. Identifying these components helps in visualizing a circle even without directly plotting the graph.

Graphing Functions

Graphing functions involves plotting their points on the coordinate plane to visually represent their behavior. When we have an equation such as \(x^2 + y^2 = 100\) which is expressed in terms of \(y\), it results in two distinct functions:

• \(y_1 = \sqrt{100 - x^2}\)
• \(y_2 = -\sqrt{100 - x^2}\)

These separate functions correspond to the upper and lower halves of the circle, respectively.
The "plus-minus" operation provides two solutions or functions that, when combined, depict the complete circle.
For graphing, both \(y_1\) and \(y_2\) are plotted within the same axes, creating the full circular shape. This division into two halves not only aids in accurate graphing but also showcases the two-fold nature of solving squared terms in functions.

Symmetry in Graphs

Symmetry plays a fundamental role in recognizing and graphing shapes like circles. In the context of graph equations, symmetry refers to consistent patterns that make understanding and plotting simpler.
Graphing a circle as demonstrated with \(x^2 + y^2 = 100\) highlights a key form of symmetry: **circular symmetry**, where the figure is identical in all directions around its center.

• There is no single "axis of symmetry" in a circle as seen in parabolas, since every radius from the origin is of equal length.
• The graph reflects symmetry across both axes and through the center. Regardless of how you view it, each opposing side mirrors the other.

This characteristic of symmetry allows simpler calculations and graphing techniques, as recognizing these properties helps predict behavior, ensure balance, and support easier plotting on graphing utilities.