Which Of The Following Is A Result Of Shifting A Circle With The Equation \[$(x+3)^2+(y-2)^2=36\$\] Up 3 Units?A. Both The \[$x\$\]- And \[$y\$\]-coordinates Of The Center Of The Circle Increase By 3.B. The

Introduction to Circle Equations

A circle is a set of points in a plane that are equidistant from a central point known as the center. The equation of a circle in standard form is given by ${(x-h)^2 + (y-k)^2 = r^2}$, where ${(h, k)}$ represents the coordinates of the center of the circle, and ${r}$ is the radius. In this article, we will focus on the equation ${(x+3)^2 + (y-2)^2 = 36}$, which represents a circle with a center at ${(-3, 2)}$ and a radius of ${6}$.

Understanding Vertical Translation

When a circle is shifted up by a certain number of units, it means that the center of the circle is moved vertically upwards. This type of translation is known as a vertical translation. In this case, we are asked to shift the circle up by 3 units.

Shifting the Circle Up 3 Units

To shift the circle up 3 units, we need to add 3 to the ${y}$-coordinate of the center of the circle. The new equation of the circle will be ${(x+3)^2 + (y-2+3)^2 = 36}$. Simplifying this equation, we get ${(x+3)^2 + (y+1)^2 = 36}$.

Analyzing the Effects of Vertical Translation

Now that we have the new equation of the circle, let's analyze the effects of shifting the circle up 3 units. The center of the circle has moved from ${(-3, 2)}$ to ${(-3, 1)}$. This means that the ${y}$-coordinate of the center of the circle has decreased by 1 unit, not increased by 3 units.

Conclusion

In conclusion, shifting a circle with the equation ${(x+3)^2 + (y-2)^2 = 36}$ up 3 units results in a decrease of 1 unit in the ${y}$-coordinate of the center of the circle, not an increase of 3 units. Therefore, the correct answer is:

A. The ${y}$-coordinate of the center of the circle decreases by 1 unit, not increases by 3 units.

Understanding the Effects of Vertical Translation on Circle Equations

When a circle is shifted up by a certain number of units, the equation of the circle changes. The new equation of the circle can be obtained by adding the number of units to the ${y}$-coordinate of the center of the circle. In this case, we added 3 to the ${y}$-coordinate of the center of the circle, resulting in a new equation of ${(x+3)^2 + (y+1)^2 = 36}$.

Analyzing the Effects of Vertical Translation on Circle Properties

Shifting a circle up by a certain number of units affects the properties of the circle. The center of the circle moves vertically upwards, and the radius remains the same. In this case, the center of the circle moved from ${(-3, 2)}$ to ${(-3, 1)}$, and the radius remained the same.

Conclusion

In conclusion, shifting a circle with the equation ${(x+3)^2 + (y-2)^2 = 36}$ up 3 units results in a decrease of 1 unit in the ${y}$-coordinate of the center of the circle, not an increase of 3 units. Therefore, the correct answer is:

A. The ${y}$-coordinate of the center of the circle decreases by 1 unit, not increases by 3 units.

Real-World Applications of Circle Equations

Circle equations have many real-world applications in fields such as engineering, physics, and computer science. Understanding how to shift a circle and analyze the effects of vertical translation is crucial in these fields.

Conclusion

In conclusion, shifting a circle with the equation ${(x+3)^2 + (y-2)^2 = 36}$ up 3 units results in a decrease of 1 unit in the ${y}$-coordinate of the center of the circle, not an increase of 3 units. Therefore, the correct answer is:

A. The ${y}$-coordinate of the center of the circle decreases by 1 unit, not increases by 3 units.

Final Thoughts

Shifting a circle and analyzing the effects of vertical translation is a crucial concept in mathematics. Understanding how to shift a circle and analyze the effects of vertical translation is essential in many real-world applications. In this article, we have discussed the effects of shifting a circle with the equation ${(x+3)^2 + (y-2)^2 = 36}$ up 3 units. We have shown that the correct answer is:

A. The ${y}$-coordinate of the center of the circle decreases by 1 unit, not increases by 3 units.

References

• [1] "Circle Equations" by Math Open Reference
• [2] "Shifting a Circle" by Khan Academy
• [3] "Vertical Translation" by Wolfram MathWorld
  

Q: What is the equation of a circle in standard form?

A: The equation of a circle in standard form is given by ${(x-h)^2 + (y-k)^2 = r^2}$, where ${(h, k)}$ represents the coordinates of the center of the circle, and ${r}$ is the radius.

Q: How do I shift a circle up by a certain number of units?

A: To shift a circle up by a certain number of units, you need to add the number of units to the ${y}$-coordinate of the center of the circle. For example, if you want to shift the circle up by 3 units, you would add 3 to the ${y}$-coordinate of the center of the circle.

Q: What happens to the center of the circle when it is shifted up by a certain number of units?

A: When a circle is shifted up by a certain number of units, the center of the circle moves vertically upwards. The ${x}$-coordinate of the center of the circle remains the same, but the ${y}$-coordinate increases by the number of units.

Q: What is the effect of shifting a circle up by a certain number of units on the radius of the circle?

A: Shifting a circle up by a certain number of units does not affect the radius of the circle. The radius remains the same.

Q: How do I determine the new equation of a circle after it has been shifted up by a certain number of units?

A: To determine the new equation of a circle after it has been shifted up by a certain number of units, you need to add the number of units to the ${y}$-coordinate of the center of the circle. For example, if the original equation of the circle is ${(x+3)^2 + (y-2)^2 = 36}$ and you want to shift it up by 3 units, the new equation of the circle would be ${(x+3)^2 + (y-2+3)^2 = 36}$.

Q: What is the difference between shifting a circle up and shifting a circle down?

A: Shifting a circle up and shifting a circle down are two different types of translations. Shifting a circle up means moving the center of the circle vertically upwards, while shifting a circle down means moving the center of the circle vertically downwards.

Q: How do I shift a circle down by a certain number of units?

A: To shift a circle down by a certain number of units, you need to subtract the number of units from the ${y}$-coordinate of the center of the circle. For example, if you want to shift the circle down by 3 units, you would subtract 3 from the ${y}$-coordinate of the center of the circle.

Q: What is the effect of shifting a circle down by a certain number of units on the center of the circle?

A: When a circle is shifted down by a certain number of units, the center of the circle moves vertically downwards. The ${x}$-coordinate of the center of the circle remains the same, but the ${y}$-coordinate decreases by the number of units.

Q: Can I shift a circle by a negative number of units?

A: Yes, you can shift a circle by a negative number of units. Shifting a circle by a negative number of units is equivalent to shifting the circle in the opposite direction. For example, shifting a circle up by -3 units is equivalent to shifting the circle down by 3 units.

Q: How do I determine the new equation of a circle after it has been shifted by a negative number of units?

A: To determine the new equation of a circle after it has been shifted by a negative number of units, you need to subtract the number of units from the ${y}$-coordinate of the center of the circle. For example, if the original equation of the circle is ${(x+3)^2 + (y-2)^2 = 36}$ and you want to shift it down by 3 units, the new equation of the circle would be ${(x+3)^2 + (y-2-3)^2 = 36}$.

Q: What is the difference between shifting a circle horizontally and shifting a circle vertically?

A: Shifting a circle horizontally and shifting a circle vertically are two different types of translations. Shifting a circle horizontally means moving the center of the circle left or right, while shifting a circle vertically means moving the center of the circle up or down.

Q: How do I shift a circle horizontally by a certain number of units?

A: To shift a circle horizontally by a certain number of units, you need to add the number of units to the ${x}$-coordinate of the center of the circle. For example, if you want to shift the circle to the right by 3 units, you would add 3 to the ${x}$-coordinate of the center of the circle.

Q: What is the effect of shifting a circle horizontally by a certain number of units on the center of the circle?

A: When a circle is shifted horizontally by a certain number of units, the center of the circle moves horizontally. The ${y}$-coordinate of the center of the circle remains the same, but the ${x}$-coordinate increases or decreases by the number of units.

Q: Can I shift a circle by a combination of horizontal and vertical translations?

A: Yes, you can shift a circle by a combination of horizontal and vertical translations. For example, you can shift a circle up by 3 units and to the right by 2 units.

Q: How do I determine the new equation of a circle after it has been shifted by a combination of horizontal and vertical translations?

A: To determine the new equation of a circle after it has been shifted by a combination of horizontal and vertical translations, you need to add the number of units to the ${x}$-coordinate of the center of the circle and add the number of units to the ${y}$-coordinate of the center of the circle. For example, if the original equation of the circle is ${(x+3)^2 + (y-2)^2 = 36}$ and you want to shift it up by 3 units and to the right by 2 units, the new equation of the circle would be ${(x+3+2)^2 + (y-2+3)^2 = 36}$.