Question

sketch the graph of $(x - 4)^2+(y - 1)^2 = 9$.

Explanation:

Step1: Recall the standard form of a circle's equation

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 \((x - 4)^2+(y - 1)^2=9\), we have \(h = 4\), \(k = 1\), and \(r=\sqrt{9}=3\).

Step2: Analyze the center and radius

The center of the circle is \((4,1)\) and the radius \(r = 3\).
We need to check which of the given graphs has a center at \((4,1)\) and a radius of \(3\).

Answer:

Since the first graph (the one with the center in the second - quadrant - like position relative to the origin in the provided options) does not have center \((4,1)\), and assuming the correct graph (not fully shown in the text - based on the center \((4,1)\) and radius \(3\)) is the one that has a center at \((4,1)\) (if among the options, the one with center \((4,1)\) (4 units to the right of the \(y\) - axis and 1 unit above the \(x\) - axis) and radius 3 (distance from the center to any point on the circle is 3) is the correct choice. But since the options are not fully described, if we assume the second graph (the one with center near \(x = 4,y = 1\) in the grid) is the correct one (based on center \((4,1)\) and radius \(3\)).