Question

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

Explanation:

Step1: Identify the center and radius

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.
For the equation \((x - 4)^2+(y - 2)^2=9\), we have \(h = 4\), \(k = 2\), and \(r=\sqrt{9}=3\).

Step2: Plot the center

On the coordinate plane, mark the point \((4,2)\) as the center of the circle.

Step3: Plot points around the center

From the center \((4,2)\), move 3 units up (to \((4,2 + 3)=(4,5)\)), down (to \((4,2-3)=(4,-1)\)), left (to \((4 - 3,2)=(1,2)\)), and right (to \((4+3,2)=(7,2)\)).

Step4: Sketch the circle

Using the center \((4,2)\) and the four points \((4,5)\), \((4,-1)\), \((1,2)\), \((7,2)\) as guides, draw a smooth - curved circle.

Answer:

First, plot the center \((4,2)\). Then, from the center, move 3 units in all four cardinal directions (up, down, left, right) to get points \((4,5)\), \((4,-1)\), \((1,2)\), \((7,2)\). Finally, draw a circle passing through these points with \((4,2)\) as the center.