Problem 41 - A Graphical Approach to College Algebra

Question

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$x^{2}+y^{2}=81$$

Step-by-Step Solution

Answer

Domain: \([-9, 9]\); Range: \([-9, 9]\).

Step 1: Identify the Equation Type

The given equation is a circle equation in the form \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. For the given equation \(x^2 + y^2 = 81\), the radius \(r\) can be determined by taking the square root of 81.

Step 2: Calculate the Radius

Since the equation is \(x^2 + y^2 = r^2\), we have \(r^2 = 81\). Taking the square root of both sides, we find \(r = \sqrt{81} = 9\). So, the radius of the circle is 9.

Step 3: Determine the Center

The given circle equation \(x^2 + y^2 = 81\) is centered at the origin \((0,0)\) because the equation is in the standard form where no \(x\) or \(y\) shifts are present. Thus, \((h, k) = (0, 0)\).

Step 4: Setup the Viewing Window

When graphing the circle on a calculator, set the square viewing window such that it covers the full extent of the circle. The window should extend from \(-9\) to \(9\) for both the x-axis and y-axis, as the radius is 9 units.

Step 5: Graph the Circle

Using the graphing calculator, input the equation \(x^2 + y^2 = 81\) and use the square window settings to plot the circle centered at \((0, 0)\) with radius 9. You should see a perfect circle touching the borders of the window at \(x = \pm 9\) and \(y = \pm 9\).

Step 6: Determine the Domain

Since the circle is centered at the origin and has a radius of 9, the range of x-values (domain) that the circle covers is from \(-9\) to \(9\). Thus, the domain is \([-9, 9]\).

Step 7: Determine the Range

Similarly, the range of y-values that the circle covers is also from \(-9\) to \(9\), due to its radius. Therefore, the range is \([-9, 9]\).

Key Concepts

Graphing Calculator

Graphing calculators are essential tools for visualizing mathematical concepts, and graphing a circle is no exception. These calculators allow you to input the circle's equation and observe its representation on a coordinate plane.

When graphing a circle, it's crucial to use a square viewing window. This ensures that the circle appears as a perfect round shape, rather than an ellipse or distorted form. For example, with the equation \(x^2 + y^2 = 81\), the circle is centered at the origin \((0,0)\) and has a radius of 9. Therefore, you should set your graphing calculator window to display values from \(-9\) to \(9\) for both the x-axis and y-axis settings.

Here’s how you can set up your graphing calculator:

• Turn on your calculator and access the graphing mode.
• Input the circle's equation: \(x^2 + y^2 = 81\).
• Adjust the window settings to have a range of \(-9\) to \(9\) for both axis.
• Ensure the window is square to preserve the circle's proportions.
• Press the graph button to view the circle on the display.

Domain and Range

Understanding the domain and range of a circle is crucial for correctly interpreting its graph. The domain refers to all possible x-values that the circle's equation can take. In the case of the equation \(x^2 + y^2 = 81\), the domain is defined by the circle’s extent along the x-axis.

Since the circle is centered at the origin (0,0) and has a radius of 9, the domain covers x-values from \(-9\) to \(9\). This means you will only see parts of the circle between these boundaries horizontally.

Similarly, the range is the set of all possible y-values. The range for this circle is also from \(-9\) to \(9\), covering the vertical distance the circle spans. Both the domain and range being the same indicates the circle is perfectly centered and symmetric, without any horizontal or vertical shifts.

• Domain: \([-9, 9]\)
• Range: \([-9, 9]\)

Recognizing these limits helps in accurately setting up your graph and understanding the spatial limits of the circle on a graphing calculator.

Radius of a Circle

The radius of a circle is a fundamental measurement that helps define its size. It's the distance from the center of the circle to any point on its circumference. When given a circle equation in the form \(x^2 + y^2 = r^2\), the radius \(r\) can be determined by taking the square root of the constant on the right-hand side of the equation.

In our example with the equation \(x^2 + y^2 = 81\), defining the circle’s radius involves solving \(r^2 = 81\). By taking the square root of 81, we find that \(r = \sqrt{81} = 9\). Thus, the circle has a radius of 9. It is essential to understand how the radius affects not only the size of the circle but also how it appears on a graph.

Here's why the radius is important:

• It determines the overall size of the circle on the graph.
• Helps in setting the correct viewing window for accurate graphing.
• Is crucial in defining the domain and range for the circle.

The radius is a vital concept to grasp as it directly connects the geometry of the circle with its algebraic equation.