Question

question 18 of 25
which of the following is a result of shifting a circle with equation ((x + 3)^2 + (y - 2)^2 = 36) right 3 units?

a. the (y)-coordinate of the center of the circle decreases by 3.

b. the (x)-coordinate of the center of the circle increases by 3.

c. the (y)-coordinate of the center of the circle increases by 3.

d. both the (x)- and (y)-coordinates of the center of the circle increase by 3.

Explanation:

Step1: Recall circle equation form

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 given equation \((x + 3)^2 + (y - 2)^2 = 36\), we can rewrite it as \((x - (-3))^2 + (y - 2)^2 = 6^2\), so the center is \((-3, 2)\).

Step2: Analyze horizontal shift

Shifting a graph right 3 units affects the \(x\)-coordinate of the center. The rule for a horizontal shift right by \(a\) units is to replace \(x\) with \(x - a\) in the equation. For our center \((-3, 2)\), shifting right 3 units means we add 3 to the \(x\)-coordinate: \(-3 + 3 = 0\). The \(y\)-coordinate remains unchanged (2). So the \(x\)-coordinate of the center increases by 3.

Answer:

B. The x - coordinate of the center of the circle increases by 3.