Question

determine whether the following statement makes sense or does not make sense, and explain the reasoning.
the graph of $(x + 4)^2+(y - 3)^2 = 36$ is the graph of $x^2 + y^2 = 36$ translated four units left and three units up.
select the correct choice below and fill in the answer box to complete your choice.
○ a. the statement does not make sense because the graph of $x^2 + y^2 = r^2$ is translated $|k|$ units horizontally and $|h|$ units vertically to obtain the graph of $(x - h)^2+(y - k)^2 = r^2$, where $(h,k)$ is the center of the graph. for $(x + 4)^2+(y - 3)^2 = 36$, $(h,k)=□$.
(type an ordered pair.)
○ b. the statement makes sense because the graph of $x^2 + y^2 = r^2$ is translated $|h|$ units horizontally and $|k|$ units vertically to obtain the graph of $(x - h)^2+(y - k)^2 = r^2$, where $(h,k)$ is the center of the graph. for $(x + 4)^2+(y - 3)^2 = 36$, $(h,k)=□$.
(type an ordered pair.)

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center of the circle.
For the equation \((x + 4)^2+(y - 3)^2=36\), we can rewrite \((x + 4)\) as \((x-(- 4))\). So, comparing with \((x - h)^2+(y - k)^2=r^2\), we have \(h=-4\) and \(k = 3\).

Step2: Analyze the translation rule

The graph of \(x^{2}+y^{2}=r^{2}\) (center \((0,0)\)) is translated \(|h|\) units horizontally and \(|k|\) units vertically to get \((x - h)^2+(y - k)^2=r^{2}\). If \(h<0\), the graph moves \(|h|\) units to the left; if \(k>0\), the graph moves \(|k|\) units up.
For \(x^{2}+y^{2}=36\) (center \((0,0)\)) and \((x + 4)^2+(y - 3)^2=36\) (center \((-4,3)\)), the graph of \(x^{2}+y^{2}=36\) is translated \(|-4|\) units left and \(|3|\) units up.

Answer:

B. The statement makes sense because the graph of \(x^{2}+y^{2}=r^{2}\) is translated \(|h|\) units horizontally and \(|k|\) units vertically to obtain the graph of \((x - h)^2+(y - k)^2=r^{2}\), where \((h,k)\) is the center of the graph. For \((x + 4)^2+(y - 3)^2=36\), \((h,k)=(-4,3)\)