Question

equation. graph the circle.
a. $(x + 3)^2 + (y - 2)^2 = 9$
general form: $(x - h)^2 + (y - k)^2 = r^2$
center: $(-3, 2)$
radius = 3
$r^2 = 9$
$r = 3$
b. $(x + 4)^2 + (y - 8)^2 = 36$

Explanation:

Step1: Recall Circle Equation Form

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

Step2: Identify \(h\), \(k\), \(r^2\) for Part b

For \((x + 4)^2 + (y - 8)^2 = 36\), rewrite \(x + 4\) as \(x - (-4)\). So, \(h = -4\), \(k = 8\), and \(r^2 = 36\).

Step3: Calculate Radius \(r\)

Take the square root of \(r^2\): \(r = \sqrt{36} = 6\).

Step4: Graphing Steps

1. Plot the center \((-4, 8)\) on the coordinate plane.
2. From the center, move 6 units left, right, up, and down (since radius is 6) to find points on the circle: \((-4 + 6, 8) = (2, 8)\), \((-4 - 6, 8) = (-10, 8)\), \((-4, 8 + 6) = (-4, 14)\), \((-4, 8 - 6) = (-4, 2)\).
3. Draw a circle through these points with the center at \((-4, 8)\).

Answer:

For the circle \((x + 4)^2 + (y - 8)^2 = 36\), the center is \((-4, 8)\) and the radius is \(6\). To graph it, plot the center and use the radius to sketch the circle.