Question

which of the following is a result of shifting a circle with equation (x + 3)^2+(y - 2)^2 = 36 up 3 units?
a. both the x - and y - coordinates of the center of the circle increase by 3.
b. the y - coordinate of the center of the circle decreases by 3.
c. the x - coordinate of the center of the circle increases by 3.
d. the y - coordinate of the center of the circle increases by 3.

Explanation:

Step1: Identify the center of the original circle

The standard - form of a circle equation is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center of the circle and $r$ is the radius. For the circle $(x + 3)^2+(y - 2)^2=36$, the center is $(-3,2)$.

Step2: Apply the vertical - shift rule

Shifting a point $(x,y)$ up $k$ units changes the point to $(x,y + k)$. Here, $k = 3$. Shifting the center $(-3,2)$ up 3 units gives the new center $(-3,2 + 3)=(-3,5)$. The $x$ - coordinate of the center remains the same, and the $y$ - coordinate of the center increases by 3.

Answer:

D. The $y$-coordinate of the center of the circle increases by 3.