To determine the correct features that Vanessa should include in her graph, we need to analyze the given equation of the circle:
[tex]\[
(x-4)^2+(y+1)^2=64
\][/tex]
First, let's rewrite this equation in its standard form. The general equation of a circle is:
[tex]\[
(x-h)^2 + (y-k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle, and [tex]\(r\)[/tex] is the radius.
By comparing the given equation [tex]\((x-4)^2 + (y+1)^2 = 64\)[/tex] with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we can identify the following:
1. Center of the circle:
- The value of [tex]\(h\)[/tex] comes from [tex]\((x-4)^2\)[/tex]. Here, [tex]\(h = 4\)[/tex].
- The value of [tex]\(k\)[/tex] comes from [tex]\((y+1)^2\)[/tex]. Here, [tex]\(k = -1\)[/tex].
- Therefore, the center of the circle is [tex]\((4, -1)\)[/tex].
2. Radius of the circle:
- The right side of the equation, [tex]\(64\)[/tex], represents [tex]\(r^2\)[/tex].
- To find the radius [tex]\(r\)[/tex], we need to take the square root of [tex]\(64\)[/tex].
[tex]\[
r^2 = 64 \implies r = \sqrt{64} = 8
\][/tex]
Putting these findings together, the center of the circle is [tex]\((4, -1)\)[/tex] and the radius is [tex]\(8\)[/tex].
From the provided options:
A. a center at [tex]\((4,-1)\)[/tex] and a radius of 64
B. a center at [tex]\((-4,1)\)[/tex] and a radius of 8
C. a center at [tex]\((4,-1)\)[/tex] and a radius of 8
D. a center at [tex]\((-4,1)\)[/tex] and a radius of 64
The correct answer is:
C. a center at [tex]\((4,-1)\)[/tex] and a radius of 8
