Problem 44 - A Graphical Approach to College Algebra

Question

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$(x+2)^{2}+(y+3)^{2}=36$$

Step-by-Step Solution

Answer

Domain: [-8, 4]; Range: [-9, 3].

Step 1: Identify Circle Parameters

The given equation is \((x+2)^2 + (y+3)^2 = 36\). This is a standard form of the circle equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. For the given circle, the center \((h, k)\) is \((-2, -3)\) and the radius \(r\) is \(\sqrt{36} = 6\).

Step 2: Set up Graphing Calculator

Input the function into the graphing calculator. Since it’s a circle, ensure the calculator's viewing window is a square to maintain the aspect ratio. You can set the window, for example, from -10 to 10 for both x and y axes.

Step 3: Determine Domain and Range

The domain of the circle is the set of all possible x-values. Since the radius is 6 and the center is at \(x = -2\), the domain is from \(-2 - 6 = -8\) to \(-2 + 6 = 4\). The range is the set of all possible y-values, so from \(y = -3 - 6 = -9\) to \(y = -3 + 6 = 3\).

Key Concepts

Domain and Range of a Circle

Understanding the domain and range of a circle can greatly enhance your grasp of circle equations. The domain refers to all possible x-values that the circle can take. In the equation \((x+2)^2 + (y+3)^2 = 36\), the center is located at \((-2, -3)\) and the radius is 6. This means the circle stretches 6 units in all directions from the center. Thus, the smallest x-value is

• -2 - 6 = -8

and the largest is

• -2 + 6 = 4.

So, the domain is from \(-8\) to \(4\).
The range comprises all possible y-values. Similarly, the range will start at

• -3 - 6 = -9

and end at

• -3 + 6 = 3.

Therefore, the range of the circle is from \(-9\) to \(3\). By understanding the domain and range, you're able to identify the extent of the circle along each axis.

Graphing Calculator Tips

Using a graphing calculator can make visualizing complex equations like circles much easier. To graph a circle, input the equation \((x+2)^2 + (y+3)^2 = 36\) directly.
Ensure that your calculator's viewing window is square. This means both x and y axes need to have the same range to accurately represent a circle without distortion.
A good practice is to set both axes from

• -10 to 10.

This provides a wide enough view while keeping the circle centered and correctly proportioned. Many graphing calculators also have options to adjust settings and styles, so make sure to explore these to enhance the visualization.

Radius and Center of a Circle

The equation of a circle can look daunting, but breaking it down reveals essential information. Given \((x+2)^2 + (y+3)^2 = 36\), this formula is in the standard form \((x-h)^2 + (y-k)^2 = r^2\). Here:

• The center
  • \((h, k) = (-2, -3)\)
• The radius
  • \(r = \sqrt{36} = 6\)

The center is the point from which every point on the circle is equidistant. The radius, a constant distance from the center to any point on the circle, defines the size of the circle. Visualizing this can help in graphing and understanding the circle's properties. Knowing these can help you better understand how the circle sits on a coordinate plane.