Question

The equation of a circle  (x−5)2+(y+6)2=1  has a center of  (5,−6) . Which of the following functions correctly transforms the circle so that it is shifted 2 units to the right and 3 units up?(1 point)

Answer Options are:

(x-3)^2+(y+9)^2=1

(x-7)^2+(y+9)^2=1

(x+2)^2+(y+3)^2=1

(x-2)^2+(y-3)^2=1

Answer 1

[AI-generated answer — Bot GPT 3.5]

To shift the circle 2 units to the right and 3 units up, we need to subtract 2 from the x-coordinate and add 3 to the y-coordinate of the center.

The correct equation of the transformed circle is therefore:
(x-2)^2 + (y+3)^2 = 1

So the correct answer is:
(x-2)^2 + (y+3)^2 = 1

Answer 2

[AI-generated answer — Step-by-Step Bot]

To shift the circle 2 units to the right and 3 units up, we need to adjust the (x, y) coordinates of the center by adding 2 to the x-coordinate and adding 3 to the y-coordinate.

Therefore, the correct transformation of the circle is:
(x - 5 + 2)^2 + (y + 6 + 3)^2 = 1

Simplifying, we get:
(x - 3)^2 + (y + 9)^2 = 1

So, the correct answer is:
(x - 3)^2 + (y + 9)^2 = 1

Answer 3

[AI-generated answer — Explain Bot]

To shift a circle 2 units to the right and 3 units up, we need to adjust the coordinates of its center accordingly.

The original center is (5, -6). To shift it 2 units to the right, we add 2 to the x-coordinate, resulting in a new x-coordinate of 5 + 2 = 7. To shift it 3 units up, we add 3 to the y-coordinate, giving us a new y-coordinate of -6 + 3 = -3.

Therefore, the new center of the shifted circle is (7, -3).

Now let's examine the answer options to see which one represents the shifted circle correctly.

Option 1: (x-3)^2+(y+9)^2=1
This option does not shift the circle correctly, as the x-coordinate of the center is shifted to the left instead of 2 units to the right.

Option 2: (x-7)^2+(y+9)^2=1
This option correctly shifts the circle 2 units to the right, as the x-coordinate of the center is now 7. However, it does not shift the circle 3 units up, as the y-coordinate of the center is still 9 units above the original center.

Option 3: (x+2)^2+(y+3)^2=1
This option does not shift the circle correctly, as the new center is shifted 2 units to the left instead of 2 units to the right.

Option 4: (x-2)^2+(y-3)^2=1
This option correctly shifts the circle 2 units to the right, as the x-coordinate of the center is now 2. It also correctly shifts the circle 3 units up, as the y-coordinate of the center is now -3.

Therefore, the correct answer is option 4: (x-2)^2+(y-3)^2=1.