Problem 69 - College Algebra

Question

Use a graphing utility to graph each circle whose equation is given. $$ (y+1)^{2}=36-(x-3)^{2} $$

Step-by-Step Solution

Answer

The circle with the equation \((y+1)^{2}=36-(x-3)^{2}\) has the center at (-1, 3) and radius 6 units. The graph of this circle can be sketched using a graphing utility by first locating the center at (-1, 3) and then drawing a circle with radius 6 units from this center point.

Step 1: Identifying the Circle Parameters

The given equation can be identified as a circle equation. The general form of the equation of a circle is \((y-h)^{2} = r^{2} - (x-k)^{2}\). By comparing this with the given equation, we can identify that the center (h, k) is at the point (-1, 3) and the radius squared (\(r^{2}\)) is 36.

Step 2: Calculating the Radius

Having the radius squared, we need to square root it to find the actual radius. Therefore, \(r = \sqrt{36} = 6\)

Step 3: Graphing the Circle

For graphing, first locate the center of the circle at point (-1, 3). From the center, mark off the radius length (6 units) in all directions. Draw a circle that passes through all these marked points. This circle is the graph of the given circle equation.