Write the standard form of the equation how to graph if the center is (2,-3) and radius is 6

To write the equation of a circle in standard form and understand how to graph it given the center and radius, you can follow these steps:

### Equation of a Circle in Standard Form

The standard form of a circle's equation is:

\[
(x - h)^2 + (y - k)^2 = r^2
\]

where:
- \((h, k)\) is the center of the circle.
- \(r\) is the radius of the circle.

### Given Data

- **Center**: \((h, k) = (2, -3)\)
- **Radius**: \(r = 6\)

### Step 1: Substitute the Given Values

Substitute the values of the center and radius into the standard form equation:

\[
(x - 2)^2 + (y + 3)^2 = 6^2
\]

### Step 2: Simplify the Equation

Calculate the square of the radius:

\[
(x - 2)^2 + (y + 3)^2 = 36
\]

So, the standard form of the equation of the circle is:

\[
(x - 2)^2 + (y + 3)^2 = 36
\]

### How to Graph the Circle

1. **Plot the Center**: Mark the point \((2, -3)\) on the coordinate plane. This point is the center of the circle.

2. **Draw the Radius**: From the center, measure a distance of 6 units in all directions (up, down, left, and right) to outline the circumference of the circle.

3. **Sketch the Circle**: Draw a smooth curve connecting these points to form the circle.

### Example Graph

Here's a visual representation:

```plaintext
     y
     ^
  3  |                      x
     |                     /  
  2  |                 o 
     |                
  1  |                    
     |                 
     |                
 -3--+----1----2----o--4----5-----> x
     |             (2, -3)
     |
-1   | 
     |
```

### Key Points on the Circle

Since the radius is 6, the key points you should mark (besides the center) are:

- To the right: \((8, -3)\)
- To the left: \((-4, -3)\)
- Above: \((2, 3)\)
- Below: \((2, -9)\)

### Summary

The equation of the circle with center \((2, -3)\) and radius \(6\) is:

\[
(x - 2)^2 + (y + 3)^2 = 36
\]

### Questions

1. **How do you find the equation of a circle if given three points on its circumference?**
2. **What happens to the circle's graph if the center is moved to the origin \((0, 0)\)?**
3. **How would the equation change if the circle were translated left by 3 units and up by 2 units?**
4. **How can you derive the circle's equation if given the endpoints of its diameter instead of the center and radius?**
5. **What are the similarities and differences between a circle's equation in standard form and in general form?**
6. **How do you graph a circle given in general form: \(x^2 + y^2 + Dx + Ey + F = 0\)?**
7. **How do you convert a circle's equation from standard form to general form?**
8. **How can you use the distance formula to verify points that lie on the circle's circumference?**

### Tip

To easily verify if a point \((x, y)\) lies on the circle, substitute it into the circle's equation. If the equation holds true, then the point is on the circle.

Learn how to write the standard form of the equation and graph a circle with center at (2, -3) and radius 6. This problem covers key geometric concepts and the standard form of circle equations. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation