Shifting a Circle Up 3 Units

Oct 13, 2025 by miodeya.com 65 views

Introduction

In mathematics, a circle is a set of points that are all equidistant from a central point called the center. The equation of a circle is a fundamental concept in geometry, and it is used to describe the shape and position of a circle in a coordinate plane. In this article, we will explore the equation of a circle and how it changes when the circle is shifted up 3 units.

The Equation of a Circle

The general equation of a circle with center ${(h, k)}$ and radius ${r}$ is given by:

${(x - h)^2 + (y - k)^2 = r^2}$

This equation describes a circle with center ${(h, k)}$ and radius ${r}$ . The center of the circle is the point ${(h, k)}$ , and the radius is the distance from the center to any point on the circle.

Shifting a Circle Up 3 Units

When a circle is shifted up 3 units, its center is moved up 3 units, and its equation changes accordingly. To shift a circle up 3 units, we need to add 3 to the ${y}$ -coordinate of the center.

The New Equation of the Circle

The original equation of the circle is:

${(x+3)^2+(y-2)^2=36}$

To shift the circle up 3 units, we need to add 3 to the ${y}$ -coordinate of the center. This gives us a new equation:

${(x+3)^2+(y-2+3)^2=36}$

Simplifying the equation, we get:

${(x+3)^2+(y+1)^2=36}$

Analyzing the New Equation

The new equation of the circle is:

${(x+3)^2+(y+1)^2=36}$

Comparing this equation with the original equation, we can see that the ${x}$ -coordinate of the center remains the same, but the ${y}$ -coordinate of the center has changed. The new center of the circle is ${(-3, -1)}$ .

Conclusion

In conclusion, when a circle is shifted up 3 units, its center is moved up 3 units, and its equation changes accordingly. The new equation of the circle is obtained by adding 3 to the ${y}$ -coordinate of the center. The ${x}$ -coordinate of the center remains the same, but the ${y}$ -coordinate of the center changes.

Answer

Based on the analysis above, the correct answer is:

The ${$ y$}$-coordinate of the center of the circle increases by 3.

Discussion

The equation of a circle is a fundamental concept in geometry, and it is used to describe the shape and position of a circle in a coordinate plane. When a circle is shifted up 3 units, its center is moved up 3 units, and its equation changes accordingly. The new equation of the circle is obtained by adding 3 to the ${y}$ -coordinate of the center.

Example

Suppose we have a circle with the equation:

${(x-2)^2+(y+4)^2=16}$

To shift this circle up 3 units, we need to add 3 to the ${y}$ -coordinate of the center. This gives us a new equation:

${(x-2)^2+(y+4+3)^2=16}$

Simplifying the equation, we get:

${(x-2)^2+(y+7)^2=16}$

The new center of the circle is ${(2, -7)}$ .

Tips and Tricks

When shifting a circle up 3 units, remember to add 3 to the ${y}$ -coordinate of the center. This will give you the new equation of the circle.

Common Mistakes

When shifting a circle up 3 units, be careful not to change the ${x}$ -coordinate of the center. The ${x}$ -coordinate of the center remains the same, but the ${y}$ -coordinate of the center changes.

Conclusion

Q: What is the equation of a circle?

A: The general equation of a circle with center ${(h, k)}$ and radius ${r}$ is given by:

${(x - h)^2 + (y - k)^2 = r^2}$

Q: How do I shift a circle up 3 units?

A: To shift a circle up 3 units, you need to add 3 to the ${y}$ -coordinate of the center. This will give you the new equation of the circle.

Q: What happens to the ${x}$ -coordinate of the center when shifting a circle up 3 units?

A: The ${x}$ -coordinate of the center remains the same when shifting a circle up 3 units. Only the ${y}$ -coordinate of the center changes.

Q: How do I find the new equation of a circle after shifting it up 3 units?

A: To find the new equation of a circle after shifting it up 3 units, you need to add 3 to the ${y}$ -coordinate of the center. This will give you the new equation of the circle.

Q: What is the new equation of a circle with the equation ${(x+3)^2+(y-2)^2=36}$ after shifting it up 3 units?

A: The new equation of the circle after shifting it up 3 units is:

${(x+3)^2+(y-2+3)^2=36}$

Simplifying the equation, we get:

${(x+3)^2+(y+1)^2=36}$

Q: What is the center of the circle after shifting it up 3 units?

A: The center of the circle after shifting it up 3 units is ${(-3, -1)}$ .

Q: How do I shift a circle down 3 units?

A: To shift a circle down 3 units, you need to subtract 3 from the ${y}$ -coordinate of the center. This will give you the new equation of the circle.

Q: What happens to the ${x}$ -coordinate of the center when shifting a circle down 3 units?

A: The ${x}$ -coordinate of the center remains the same when shifting a circle down 3 units. Only the ${y}$ -coordinate of the center changes.

Q: How do I find the new equation of a circle after shifting it down 3 units?

A: To find the new equation of a circle after shifting it down 3 units, you need to subtract 3 from the ${y}$ -coordinate of the center. This will give you the new equation of the circle.

Q: What is the new equation of a circle with the equation ${(x-2)^2+(y+4)^2=16}$ after shifting it down 3 units?

A: The new equation of the circle after shifting it down 3 units is:

${(x-2)^2+(y+4-3)^2=16}$

Simplifying the equation, we get:

${(x-2)^2+(y+1)^2=16}$

Q: What is the center of the circle after shifting it down 3 units?

A: The center of the circle after shifting it down 3 units is ${(2, 1)}$ .

Conclusion

In conclusion, shifting a circle up or down 3 units changes its equation accordingly. The new equation of the circle is obtained by adding or subtracting 3 from the ${y}$ -coordinate of the center. The ${x}$ -coordinate of the center remains the same, but the ${y}$ -coordinate of the center changes.