Question

question determine the equation of the circle graphed below.

Explanation:

Step1: Identify the center of the circle

From the graph, we can see that the center of the circle \((h,k)\) is at \((-1, 4)\). This is determined by looking at the coordinates where the center lies on the coordinate plane.

Step2: Determine the radius of the circle

To find the radius, we can count the number of units from the center to a point on the circle. For example, from the center \((-1, 4)\) to a point on the circle, say when \(x = -1\) and \(y = 10\) (or other points), we calculate the distance. The radius \(r\) can also be found by looking at the horizontal or vertical distance. From the center \((-1,4)\) to the rightmost point (or leftmost, topmost, bottommost), we can see that the radius is \(5\) (by counting the grid units). We can verify this using the distance formula, but visually, it's clear that the radius is \(5\).

Step3: Use the standard equation of a circle

The standard equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius. Substituting \(h=-1\), \(k = 4\) and \(r = 5\) into the equation, we get:

$$ (x - (-1))^2 + (y - 4)^2 = 5^2 $$

Simplifying the left - hand side and the right - hand side:

$$ (x + 1)^2 + (y - 4)^2 = 25 $$

Answer:

The equation of the circle is \((x + 1)^2+(y - 4)^2 = 25\)