Problem 42 - A Graphical Approach to College Algebra

Question

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$x^{2}+(y+3)^{2}=49$$

Step-by-Step Solution

Answer

Domain: [-7, 7]; Range: [-10, 4]

Step 1: Understanding the Equation of a Circle

The given equation is in the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\). Here, \(h = 0\), \(k = -3\), and \(r^2 = 49\) means the circle is centered at \((0, -3)\) with a radius \(r = 7\).

Step 2: Setting the Viewing Window

When using a graphing calculator, select a square viewing window to ensure the proportions are correct. Set the x-axis and y-axis to range from \(-10\) to \(10\) so that the full circle can be viewed, given that the radius is \(7\).

Step 3: Graphing the Circle

Input the equation \(x^2 + (y + 3)^2 = 49\) into the graphing calculator. The circle will be drawn with its center at \((0, -3)\) and passing through points such as \((0, 4)\) and \((0, -10)\).

Step 4: Determining the Domain

The domain of the circle is the set of all possible x-values. Since the radius is \(7\) and the circle is centered at \((0, -3)\), the x-values range from \(-7\) to \(7\). Thus, the domain is \([-7, 7]\).

Step 5: Determining the Range

The range of the circle is the set of all possible y-values. Since the center is \((0, -3)\) and the radius is \(7\), the y-values range from \(-10\) to \(4\). Thus, the range is \([-10, 4]\).

Key Concepts

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical equations and graphs. When graphing a circle, using a graphing calculator can simplify the process and make it more accurate. Begin by entering the equation of the circle in the provided input area. This equation will showcase a circle centered at a specific point with a particular radius.
Choosing the right viewing window is crucial. For the equation given, \(x^2 + (y + 3)^2 = 49\), set both the x-axis and y-axis from \(-10\) to \(10\). This ensures the complete circle is visible on the screen.

• Input the circle equation accurately to avoid graphing errors.
• Select a square viewing window for correct aspect ratio.
• Identify key points, such as the center and intersection with axes, for better understanding.

Using a graphing calculator helps in visually confirming that the drawing aligns with expected results, enhancing understanding of the circle’s geometry.

Domain and Range

Understanding the domain and range of a circle is essential for fully grasping its properties on a coordinate plane. The domain refers to all possible x-values that the circle covers, while the range refers to all possible y-values.
For a circle centered at \((0, -3)\) with a radius \(r = 7\), the domain and range can be determined as follows:

• Domain: Since the circle's radius is \(7\), it stretches from \(-7\) to \(7\) along the x-axis. The domain is \([-7, 7]\).
• Range: Vertically, the circle spans from \(-10\) to \(4\) along the y-axis. The range is \([-10, 4]\).

Grasping these values helps visualize where the circle lies in the coordinate system. Each pair of (x, y) within these limits will satisfy the equation of the circle.

Standard Form of a Circle

The standard form equation of a circle reveals essential information about its characteristics. The equation \((x - h)^2 + (y - k)^2 = r^2\) helps identify the circle's center and radius easily.
For the given example \(x^2 + (y + 3)^2 = 49\):

• Center (h, k): The equation indicates a center at \((0, -3)\). The values of \(h\) and \(k\) are derived from the transformations applied to \(x\) and \(y\).
• Radius (r): Solving \(r^2 = 49\) yields \(r = 7\). This radius dictates the size of the circle.

Recognizing the standard form allows you to immediately sketch the basic outline of the circle, such as where it is centered and its overall size on a coordinate grid. This form is fundamental in connecting algebraic expressions with geometric figures.