Question

1. graph the equation of the circle: ((x - 4)^2 + (y - 2)^2 = 9)

Explanation:

Step1: Recall the circle equation formula

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Step2: Identify center and radius

For the equation \((x - 4)^2 + (y - 2)^2 = 9\), we can rewrite \(9\) as \(3^2\). So, comparing with the standard form, \(h = 4\), \(k = 2\), and \(r = 3\). This means the center of the circle is at the point \((4, 2)\) and the radius is \(3\) units.

Step3: Plot the center

On the given coordinate grid, find the point where \(x = 4\) and \(y = 2\) and mark it as the center.

Step4: Draw the circle

Using the center \((4, 2)\) and a radius of \(3\) (so we can move \(3\) units left, right, up, and down from the center to find points on the circle: left to \(4 - 3 = 1\), right to \(4 + 3 = 7\), up to \(2 + 3 = 5\), down to \(2 - 3 = -1\)), then draw a circle passing through these points (and all other points \(3\) units from the center) on the grid.

Answer:

To graph the circle \((x - 4)^2 + (y - 2)^2 = 9\):

1. Center: Plot the point \((4, 2)\) (since \(h = 4\), \(k = 2\) from the standard circle equation \((x - h)^2 + (y - k)^2 = r^2\)).
2. Radius: The radius \(r = 3\) (since \(r^2 = 9\), so \(r = 3\)). From the center \((4, 2)\), move \(3\) units left (\(x = 1\), \(y = 2\)), right (\(x = 7\), \(y = 2\)), up (\(x = 4\), \(y = 5\)), and down (\(x = 4\), \(y = -1\)) to identify points on the circle. Then draw a circle through these (and all equidistant) points.

(Note: On the provided grid, the center \((4, 2)\) lies in the first quadrant, 4 units right of the y - axis and 2 units above the x - axis. The circle will extend from \(x = 1\) to \(x = 7\) and \(y=-1\) to \(y = 5\) relative to this center.)