Question

determine whether the following statement makes sense or does not make sense, and explain the reasoning. the graph of $(x + 4)^2+(y + 2)^2 = 49$ is the graph of $x^2+y^2 = 49$ translated four units left and two units down. select the correct choice below and fill in the answer box to complete your choice. a. 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 $(x + 4)^2+(y + 2)^2 = 49$, $(h,k)=$ (type an ordered pair.) b. 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 $(x + 4)^2+(y + 2)^2 = 49$, $(h,k)=$ (type an ordered pair.)

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 of the circle.
For the equation \((x + 4)^{2}+(y + 2)^{2}=49\), we can rewrite it as \((x-(- 4))^{2}+(y-(-2))^{2}=7^{2}\).

Step2: Determine the center of the circle

Comparing with the standard form \((x - h)^{2}+(y - k)^{2}=r^{2}\), we have \(h=-4\) and \(k = - 2\). So the center \((h,k)=(-4,-2)\).
The graph of \(x^{2}+y^{2}=r^{2}\) (center \((0,0)\)) is translated \(|h| = |-4|=4\) units horizontally (left since \(h=-4<0\)) and \(|k|=|-2| = 2\) units vertically (down since \(k=-2<0\)) to get the graph of \((x + 4)^{2}+(y + 2)^{2}=49\) (center \((-4,-2)\)).

Answer:

A. 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 \((x + 4)^{2}+(y + 2)^{2}=49\), \((h,k)=(-4,-2)\)