Sure, let's solve this step by step.
We start with the given equation of the circle:
[tex]\[
(x + 3)^2 + (y - 4)^2 = 16
\][/tex]
### Step 1: Identify the center of the circle
The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is its radius.
In our equation [tex]\((x + 3)^2 + (y - 4)^2 = 16\)[/tex], we need to rewrite it in the standard form. Notice that:
- The term [tex]\((x + 3)^2\)[/tex] can be written as [tex]\((x - (-3))^2\)[/tex] which tells us [tex]\(h = -3\)[/tex].
- The term [tex]\((y - 4)^2\)[/tex] already matches the standard form indicating [tex]\(k = 4\)[/tex].
Therefore, the center [tex]\((h, k)\)[/tex] of the circle is:
[tex]\[
(h, k) = (-3, 4)
\][/tex]
### Step 2: Identify the radius of the circle
The right-hand side of the given equation [tex]\((x + 3)^2 + (y - 4)^2 = 16\)[/tex] is [tex]\(16\)[/tex], and this corresponds to [tex]\(r^2\)[/tex] in the standard form.
To find the radius [tex]\(r\)[/tex], we take the square root of [tex]\(16\)[/tex]:
[tex]\[
r = \sqrt{16} = 4
\][/tex]
### Step 3: Summarize the findings
- The center of the circle is [tex]\((-3, 4)\)[/tex]
- The radius of the circle is [tex]\(4\)[/tex]
### Step 4: Graph the circle
To graph the circle, follow these steps:
1. Plot the center of the circle at [tex]\((-3, 4)\)[/tex] on the coordinate plane.
2. Use the radius of [tex]\(4\)[/tex] to draw a circle around the center. This means the circle will be [tex]\(4\)[/tex] units away from the center in all directions (left, right, up, and down).
Here’s how you can graphically represent it on the coordinate plane:
1. Start at the point [tex]\((-3, 4)\)[/tex].
2. Mark points that are [tex]\(4\)[/tex] units away from the center on all sides: [tex]\((-7, 4)\)[/tex], [tex]\((1, 4)\)[/tex], [tex]\((-3, 8)\)[/tex], and [tex]\((-3, 0)\)[/tex].
3. Draw a smooth curve to connect these points, forming a circle.
This is the way to approach solving and graphing your circle.
