Sure, let's go through a detailed, step-by-step solution for understanding and graphing the given equation [tex]\((x+4)^2 + (y-7)^2 = 49\)[/tex].
### Step 1: Recognize the Equation
The given equation [tex]\((x+4)^2 + (y-7)^2 = 49\)[/tex] is in the standard form for the equation of a circle.
### Step 2: Identify the Standard Form
The standard form of a circle's equation is [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], where:
- [tex]\((h, k)\)[/tex] is the center of the circle.
- [tex]\(r\)[/tex] is the radius of the circle.
### Step 3: Compare and Extract Parameters
From comparing the given equation [tex]\((x+4)^2 + (y-7)^2 = 49\)[/tex] with the standard form [tex]\((x-h)^2 + (y-k)^2 = r^2\)[/tex], we can extract the parameters:
- The center [tex]\((h, k)\)[/tex] of the circle is:
- [tex]\(h = -4\)[/tex] (since [tex]\((x+4)\)[/tex] means [tex]\(x - (-4)\)[/tex])
- [tex]\(k = 7\)[/tex] (since [tex]\((y-7)\)[/tex] means [tex]\(y - 7\)[/tex])
- The radius squared ([tex]\(r^2\)[/tex]) is 49. Therefore, the radius ([tex]\(r\)[/tex]) is:
- [tex]\(r = \sqrt{49} = 7\)[/tex]
### Step 4: Conclusion
Given these parameters, we now know that:
- The center of the circle is at [tex]\((-4, 7)\)[/tex].
- The radius of the circle is [tex]\(7\)[/tex].
So, the graph of the equation [tex]\((x+4)^2 + (y-7)^2 = 49\)[/tex] is a circle centered at the point [tex]\((-4, 7)\)[/tex] with a radius of [tex]\(7\)[/tex].
