Answer :
To draw a circle with center [tex]\((2, -3)\)[/tex] and a radius of [tex]\(3\)[/tex], let's follow the steps below:
### Step 1: Understand the Circle Equation
The general equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
For our case:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = -3\)[/tex]
- [tex]\(r = 3\)[/tex]
So, the equation of the circle is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 3^2 \][/tex]
Which simplifies to:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 9 \][/tex]
### Step 2: Plot the Center of the Circle
First, we mark the center of the circle at point [tex]\((2, -3)\)[/tex].
### Step 3: Determine Points on the Circle
To draw the circle, we need to determine several points that lie on the circumference. Here are a few key points we can plot based on the radius of 3 units from the center:
1. Rightmost point: [tex]\((2 + 3, -3) = (5, -3)\)[/tex]
2. Leftmost point: [tex]\((2 - 3, -3) = (-1, -3)\)[/tex]
3. Topmost point: [tex]\((2, -3 + 3) = (2, 0)\)[/tex]
4. Bottommost point: [tex]\((2, -3 - 3) = (2, -6)\)[/tex]
### Step 4: Draw the Circle
Using a compass, place the point at the center [tex]\((2, -3)\)[/tex] and adjust the compass to a 3-unit radius. Draw the circle starting from the center.
If you're drawing this on graph paper or using software:
- Ensure your x and y axes are scaled equally for an accurate representation.
- Plot the key points mentioned and sketch the curve connecting these points, forming the circle.
### Visual Representation
Here is the visual representation including the axes, grid, and the circle with the mentioned points:
1. Plot the Center: [tex]\( (2, -3) \)[/tex]
2. Plot Key Points: [tex]\( (5, -3), (-1, -3), (2, 0), (2, -6) \)[/tex]
3. Draw the circle through these points, ensuring a smooth curve.
### Final Check:
Ensure that:
- The curve is smooth and rounded, intersecting all key points properly.
- The center is clearly marked.
- The radius is consistent at every point from the center.
Your graph should now have a circle centered at [tex]\((2, -3)\)[/tex] with radius [tex]\(3\)[/tex].
### Step 1: Understand the Circle Equation
The general equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
For our case:
- [tex]\(h = 2\)[/tex]
- [tex]\(k = -3\)[/tex]
- [tex]\(r = 3\)[/tex]
So, the equation of the circle is:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 3^2 \][/tex]
Which simplifies to:
[tex]\[ (x - 2)^2 + (y + 3)^2 = 9 \][/tex]
### Step 2: Plot the Center of the Circle
First, we mark the center of the circle at point [tex]\((2, -3)\)[/tex].
### Step 3: Determine Points on the Circle
To draw the circle, we need to determine several points that lie on the circumference. Here are a few key points we can plot based on the radius of 3 units from the center:
1. Rightmost point: [tex]\((2 + 3, -3) = (5, -3)\)[/tex]
2. Leftmost point: [tex]\((2 - 3, -3) = (-1, -3)\)[/tex]
3. Topmost point: [tex]\((2, -3 + 3) = (2, 0)\)[/tex]
4. Bottommost point: [tex]\((2, -3 - 3) = (2, -6)\)[/tex]
### Step 4: Draw the Circle
Using a compass, place the point at the center [tex]\((2, -3)\)[/tex] and adjust the compass to a 3-unit radius. Draw the circle starting from the center.
If you're drawing this on graph paper or using software:
- Ensure your x and y axes are scaled equally for an accurate representation.
- Plot the key points mentioned and sketch the curve connecting these points, forming the circle.
### Visual Representation
Here is the visual representation including the axes, grid, and the circle with the mentioned points:
1. Plot the Center: [tex]\( (2, -3) \)[/tex]
2. Plot Key Points: [tex]\( (5, -3), (-1, -3), (2, 0), (2, -6) \)[/tex]
3. Draw the circle through these points, ensuring a smooth curve.
### Final Check:
Ensure that:
- The curve is smooth and rounded, intersecting all key points properly.
- The center is clearly marked.
- The radius is consistent at every point from the center.
Your graph should now have a circle centered at [tex]\((2, -3)\)[/tex] with radius [tex]\(3\)[/tex].
