Question

1. write an equation for the circle whose graph is shown.

options:
$(x + 4)^2 + (y - 2)^2 = 5$
$(x + 4)^2 + (y - 2)^2 = 25$
$(x - 4)^2 + (y - 2)^2 = 5$
$(x - 4)^2 + (y + 2)^2 = 25$

Explanation:

Step1: Identify the center of the circle

The center of the circle is at the point \((h, k)\). From the graph, the center is at \((4, -2)\)? Wait, no, looking at the grid, the blue dot (center) is at \(x = 4\), \(y=-2\)? Wait, no, the grid lines: the center is at (4, -2)? Wait, no, the options have \((x - 4)^2+(y + 2)^2 = 25\) or others. Wait, let's re-examine. The center: looking at the graph, the center is at (4, -2)? Wait, no, the y-coordinate: the center is at (4, -2)? Wait, the options: let's recall the standard equation of a circle: \((x - h)^2+(y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

Looking at the graph, the center is at (4, -2)? Wait, no, the blue dot is at (4, -2)? Wait, the y-axis: the center is at y = -2? Wait, the options: let's check the center coordinates. Let's see the options:

Option 1: \((x + 4)^2+(y - 2)^2 = 5\) → center (-4, 2)

Option 2: \((x + 4)^2+(y - 2)^2 = 25\) → center (-4, 2)

Option 3: \((x - 4)^2+(y - 2)^2 = 5\) → center (4, 2)

Option 4: \((x - 4)^2+(y + 2)^2 = 25\) → center (4, -2)

Wait, maybe I misread the center. Let's look at the graph again. The circle is drawn with center at (4, -2)? Wait, the y-coordinate: the center is at (4, -2). Then the radius: from center to the edge. Let's see the distance from center (4, -2) to, say, the point (4, 3)? No, wait, the circle goes from x=0 (left) to x=8 (right), so the diameter is 8 units? Wait, from x=0 to x=8, so radius is 5? Wait, the distance from center (4, -2) to x=9? No, the graph: the circle's leftmost point is at x= -1? Wait, no, the grid: each square is 1 unit. The center is at (4, -2). The radius: from center (4, -2) to, say, (4, 3) is 5 units? Wait, 3 - (-2) = 5? No, 3 - (-2) is 5? Wait, -2 to 3 is 5 units? Wait, no, the radius: let's count the grid squares. From center (4, -2) to the top of the circle: how many units? The top of the circle is at y=3? Wait, no, the graph shows the circle going up to y=2? Wait, maybe I made a mistake. Wait, the options: let's check the radius. The standard equation: \((x - h)^2+(y - k)^2 = r^2\). Let's check the center first.

Looking at the graph, the center is at (4, -2)? Wait, the blue dot is at (4, -2)? Wait, the y-axis: the center is at y = -2. So the center is (4, -2). Then the radius: let's see the distance from (4, -2) to (9, -2)? No, the circle's rightmost point is at x=9? Wait, the grid goes to x=10. Wait, the center is (4, -2), and the radius: from (4, -2) to (4, 3) is 5 units (since -2 to 3 is 5). Wait, 3 - (-2) = 5. So radius is 5, so \(r^2 = 25\). So the equation is \((x - 4)^2+(y + 2)^2 = 25\), which is option 4. Wait, but let's check the options again. Wait, the options:

Option 4: \((x - 4)^2+(y + 2)^2 = 25\) → center (4, -2), radius 5 (since \(r^2 =25\), so r=5). That matches.

Wait, but let's confirm the center. The blue dot is at (4, -2)? Let's see the graph: the center is at x=4, y=-2. Yes. So the standard equation is \((x - h)^2+(y - k)^2 = r^2\), where h=4, k=-2, so \((x - 4)^2+(y - (-2))^2 = r^2\) → \((x - 4)^2+(y + 2)^2 = r^2\). Then the radius: from center (4, -2) to the top of the circle: let's see the top of the circle is at y=3? Wait, no, the circle goes up to y=2? Wait, maybe I messed up the y-coordinate. Wait, the graph: the center is at (4, -2)? Or (4, 2)? Wait, the options have (y - 2) or (y + 2). Let's look at the graph again. The circle is drawn with the center at (4, -2)? Wait, the bottom of the circle is at y=-8, and the center is at y=-2, so the distance from center to bottom is 6? No, that can't be. Wait, maybe I made a mistake. Let's re-express:

Standard equation: \((x - h)^2 + (y - k)^2 = r^2\), center (h, k), radius r.

Looking at the graph, the center is at (4, -2) (since the blue dot is at x=4, y=-2). Then the radius: let's count the grid squares. From center (4, -2) to the leftmost point: x= -1? No, the leftmost point is at x= -1? Wait, no, the grid: each square is 1 unit. The center is at (4, -2). The leftmost point is at x= -1? No, 4 - 5 = -1? Wait, 4 - 5 = -1, so radius is 5. So from x=4 -5 = -1 to x=4 +5=9, so radius 5. Then the y-coordinate: from y= -2 -5 = -7 to y= -2 +5=3. So the circle goes from y=-7 to y=3, which matches the graph (the bottom is at y=-8? Wait, maybe the grid lines are 1 unit, so from y=-2 -5 = -7, but the graph shows down to y=-8. Maybe my count is off. Wait, the options: let's check the radius squared. The options have 5 or 25. 5 would be radius ~2.2, which is too small. 25 is radius 5, which is more reasonable. So the correct equation is \((x - 4)^2+(y + 2)^2 = 25\), which is option 4.

Step2: Verify the center and radius

- Center: (4, -2) → so \(h=4\), \(k=-2\), so in the equation, it's \((x - 4)^2 + (y - (-2))^2 = r^2\) → \((x - 4)^2 + (y + 2)^2 = r^2\).
- Radius: The distance from center (4, -2) to the edge. From the graph, the radius is 5 (since the circle spans 10 units in diameter, from x=-1 to x=9, so radius 5), so \(r^2 = 25\).

Thus, the equation is \((x - 4)^2 + (y + 2)^2 = 25\), which is option 4.

Answer:

D. \((x - 4)^2 + (y + 2)^2 = 25\) (Wait, the options are labeled with circles, but in the problem, the fourth option is \((x - 4)^2 + (y + 2)^2 = 25\), so the correct option is the fourth one, which is \((x - 4)^2 + (y + 2)^2 = 25\))