Problem 36 - Algebra and Trigonometry

Question

Graph \((x-2)^{2}+(y+1)^{2}=9\)

Step-by-Step Solution

Answer

The graph is a circle centered at (2, -1) with a radius of 3.

Step 1: Identify the Type of Graph

The given equation \((x-2)^{2}+(y+1)^{2}=9\) is in the form of the standard equation of a circle \((x-h)^{2}+(y-k)^{2}=r^{2}\).

Step 2: Determine the Center of the Circle

From the standard equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), compare and identify the center (h, k). Here, \(h = 2\) and \(k = -1\). Thus, the center of the circle is (2, -1).

Step 3: Calculate the Radius

Compare the equation \((x-2)^{2}+(y+1)^{2}=9\) to the standard form. We identify \(r^2 = 9\), so the radius \(r = \sqrt{9} = 3\).

Step 4: Plot the Graph

With the center at (2, -1) and a radius of 3, plot the point (2, -1) on a coordinate plane and draw a circle with a radius of 3 units around this center.

Key Concepts

standard equation of a circle

The standard equation of a circle is crucial in understanding how to graph a circle. It is generally written as \( (x-h)^{2}+(y-k)^{2}=r^{2} \). In this form:
\[(x-h)^{2}+(y-k)^{2}=r^{2}\]

Where:

• \