Problem 43 - A Graphical Approach to College Algebra

Question

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$(x-3)^{2}+(y-2)^{2}=25$$

Step-by-Step Solution

Answer

Center: (3, 2), Radius: 5; Domain: [-2, 8], Range: [-3, 7].

Step 1: Understand the Equation Format

The equation \((x-3)^2 + (y-2)^2 = 25\) represents a circle in standard form. The general equation of a circle is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify the Center and Radius

By comparing the given equation \((x-3)^2 + (y-2)^2 = 25\) to the general circle equation, we can see that the center \((h, k)\) is \((3, 2)\) and the radius \(r\) is \(5\), since \(r^2 = 25\).

Step 3: Graph the Circle

Using a graphing calculator, plot the center of the circle at \((3, 2)\). With a radius of 5, draw the circle such that all points are 5 units from the center. Ensure to use a square viewing window on your calculator to maintain the circle's proportions.

Step 4: Determine the Domain

The domain of the circle is determined by its horizontal extent. Since the center is at \(x = 3\) with a radius of 5, the circle extends 5 units to the left and right. Therefore, the domain is \([3 - 5, 3 + 5] = [-2, 8]\).

Step 5: Determine the Range

The range of the circle is determined by its vertical extent. The center is at \(y = 2\) with a radius of 5, so the circle extends 5 units up and down. Therefore, the range is \([2 - 5, 2 + 5] = [-3, 7]\).

Key Concepts

Standard Equation of a Circle

The standard equation of a circle is an essential concept in geometry. It helps describe circles using their position and size. The most common form of the equation is \((x-h)^2 + (y-k)^2 = r^2\). Here, the pair \((h, k)\) represents the center of the circle, whereas \(r\) stands for the radius.

You can think of \(h\) and \(k\) as coordinates that tell you exactly where the circle is located on a graph. The term \(r^2\) is the radius squared. It lets you know how far the circle extends from its center. Understanding this equation framework will help you graph circles accurately.

When you see an equation like \((x-3)^2 + (y-2)^2 = 25\), identify \((h, k)\) and \(r\) to visualize and graph the circle correctly. For this example, the center is \( (3,2) \) and the radius is \( \sqrt{25} = 5 \).

Radius and Center

Identifying the radius and center of a circle is the first step in graphing. In our equation \((x-3)^2 + (y-2)^2 = 25\), the center \((h, k)\) is at \((3, 2)\). These coordinates mark the exact middle of the circle on a plane.

Next, we calculate the radius. Knowing that \(r^2 = 25\), you can find the radius by taking the square root, resulting in \(r = 5\). This radius is crucial because it sets the distance from the center to any point on the circle.

Always ensure you've accurately identified these components since they guide the entire graphing process. They'll enable you to place the circle precisely on your graph and check its symmetry and proportions.

Domain and Range

After determining the center and radius, the next crucial aspect is finding the domain and range.

The **domain** of a circle is the set of possible \(x\)-values. For the circle centered at \((3, 2)\) with a radius of 5, the domain is easy to calculate. It extends 5 units to both the left and right from the center's \(x\)-coordinate of 3. Therefore, the domain becomes \([-2, 8]\).

Similarly, the **range** involves the \(y\)-values a circle can take. Since the circle stretches 5 units up and down from \(y=2\), the range results in \([-3, 7]\).

Understanding the domain and range helps to view the circle's limits and assists in setting your graphing calculator accurately.

Graphing Calculator

Using a graphing calculator can simplify the task of visualizing circles based on their equations. When inputting the circle equation \((x-3)^2 + (y-2)^2 = 25\), the calculator plots points around the center \((3, 2)\) with each being 5 units away, thus forming the circle.

It's important to adjust the calculator to a square viewing window. This ensures that the circle doesn't appear distorted and keeps its symmetry intact.

• Ensure axis units are equal.
• Set center at \((3, 2)\).
• Check horizontal and vertical grid spacing matches the circle's radius limits.

Utilizing these features can give a clear picture of how the circle looks in a graphing space, helping in precise analysis and interpretation.