Question

a circle on the coordinate plane with the quadrants labeled is shown below. the center of the circle is at point (0,2). the radius of the circle is 2 units. the following changes are made: the circle is reflected across the x - axis. the circle is moved to the right 3 units. the radius is divided by 2. which statement best represents the transformed circle? the circle is in quadrant i only. the circle is in quadrant iv only. the circle is in quadrants i and iv. the circle is in quadrants iii and iv.

Explanation:

Step1: Find the new center after reflection across the \(x\) - axis

The original center is \((0,2)\). When reflected across the \(x\) - axis, the \(y\) - coordinate changes sign. So the new center after reflection is \((0,- 2)\).

Step2: Find the new center after moving right 3 units

Using the rule \((x,y)\to(x + 3,y)\) for a horizontal translation to the right. For the center \((0,-2)\), the new center is \((0+3,-2)=(3,-2)\)

Step3: Find the new radius

The original radius \(r = 2\). After dividing the radius by \(2\), the new radius \(r'=\frac{2}{2}=1\)

The center of the circle is \((3,-2)\) and the radius is \(1\). The circle will extend from \(x = 3 - 1=2\) to \(x=3 + 1 = 4\) and from \(y=-2 - 1=-3\) to \(y=-2 + 1=-1\). Quadrant IV is the region where \(x>0\) and \(y<0\). Since the entire circle (with center \((3,-2)\) and radius \(1\)) has \(x\) - values \(2

Answer:

The circle is in quadrant IV only.