Problem 109 - A Graphical Approach to College Algebra

Question

In Exercises \(109-116\), describe the graph of the equation as either a circle or a parabola with a 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 given viewing window. $$\begin{aligned} &x^{2}+y^{2}=100\\\ &[-15,15] \text { by }[-10,10] \end{aligned}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Graph Type

The given equation is \( x^2 + y^2 = 100 \). This equation conforms to the standard form of a circle equation, \( x^2 + y^2 = r^2 \), where \( r \) is the radius. Therefore, this describes the graph of a circle.

Step 2: Determine the Radius of the Circle

Since the equation \( x^2 + y^2 = 100 \) matches the circle equation \( x^2 + y^2 = r^2 \), the radius \( r \) is the square root of 100, which is 10.

Step 3: Transform the Equation to Solve for y

To find two functions \( y_1 \) and \( y_2 \), express \( y \) in terms of \( x \). Begin by rearranging the equation: \( y^2 = 100 - x^2 \). Then take the square root of both sides to get \( y = \pm \sqrt{100 - x^2} \).

Step 4: Define Functions y1 and y2

Since \( y = \pm \sqrt{100 - x^2} \), we have two functions: \( y_1 = \sqrt{100 - x^2} \) and \( y_2 = -\sqrt{100 - x^2} \). These functions represent the upper and lower semicircles respectively.

Step 5: Graph the Functions

To graph \( y_1 \) and \( y_2 \), plot them over the given range \([-15,15] \text{ by } [-10,10]\). The plot of \( y_1 \) will be the upper semicircle and \( y_2 \) will be the lower semicircle, together forming the full circle with a radius of 10.

Key Concepts

Graph of a Circle

The circle is a fundamental geometric shape, and its graph is known as the graph of a circle. To draw this graph, you begin with the equation, which is typically given in the form of \( x^2 + y^2 = r^2 \). This particular equation represents a circle centered at the origin or the point (0,0).
Unlike other shapes, the circle hosts all its points equidistant from the center, forming a round boundary. When graphing it, you can visualize its smooth curve in either two halves: the top half, shown by the function \( y_1 = \sqrt{r^2 - x^2} \), and the bottom half portrayed by \( y_2 = -\sqrt{r^2 - x^2} \). Together, these equations provide a complete picture of a circle on a graph.

• Circle is centered at the origin.
• Graphs as a continuously round loop.
• Upper and lower halves can be expressed as separate functions.

In practicing this, use the given range for each axis in your graphing tool. In our example, with boundaries of [-15, 15] and [-10, 10], you will see the complete circular shape within these view options.

Circle Equation

A circle's equation is a mathematical way to express all points that make up the circle in a coordinate plane. The most prevalent form you'll encounter is \( x^2 + y^2 = r^2 \), where \( r \) is the radius of the circle. This standard form efficiently describes a circle, especially when centered at the origin.To rewrite this equation in a way that highlights its origin characteristics, examine \( r \) and how it connects to each component:

• \( x^2 \) and \( y^2 \): They add together representing distances in both horizontal and vertical planes.
• \( r^2 \): The square of the circle's radius, showing how wide it is.

It is essential to understand this form as it serves as a foundation for graphing and analyzing circle structures and transformations.
Keep in mind that you can modify the circle’s location by changing the equation to \((x - h)^2 + (y - k)^2 = r^2\), placing its center at \((h, k)\). This adds flexibility to describe circles situated in different positions.

Radius of a Circle

The radius is a primary characteristic of a circle, defining its size. It is the distance from the center of the circle to any point on its edge. In the standard circle equation \( x^2 + y^2 = r^2 \), \( r \) is the symbol for the radius.
To derive the radius from a given circle equation, you simply need to take the square root of the number on the right side of the equation. For example, given \( x^2 + y^2 = 100 \), the radius \( r \) equals \( \sqrt{100} \), which simplifies to 10.

• The radius is calculated by taking the square root of the constant term in the equation.
• It determines how large or small the circle appears on a graph.
• Radius provides a straightforward method to measure circular dimensions.

With the radius found, one can easily graph the circle and also perform additional calculations, such as finding the circle's circumference or area using formulae \(2\pi r\) and \(\pi r^2\), respectively.