1) Find the coordinates of the point (-1,3) after the transformation T (to the -3,2 power)

2) What are the coordinates of point (-3,4) under D (to the 3rd power)?

3) Which of the following transformations could map the point (1,2) onto the point (3,6)?
a)T (to the 2,3 power)
b) T (to the 3,2 power)
c)D (to the 3rd power)
d) r (to the y=x power)

4) Circle O is centered at the origin and has a radius of 5.
a) Write the equation for circle O.
b) Circle O' is the image of circle O after a translation of T (to the 2,-3 power). What is the center of circle O'? What is the radius of O'?
c) Write the equation for circle O'.

1) To find the coordinates of the point (-1,3) after the transformation T (to the -3,2 power), you would need to raise the x-coordinate and y-coordinate to the -3,2 power respectively.

For the x-coordinate: (-1)^(-3) = -1/(-1)^3 = -1/(-1) = 1

For the y-coordinate: (3)^2 = 3^2 = 9

Therefore, the coordinates of the transformed point would be (1,9).

2) Similarly, to find the coordinates of point (-3,4) under the transformation D (to the 3rd power), you would raise the x-coordinate and y-coordinate to the 3rd power.

For the x-coordinate: (-3)^3 = -3^3 = -27

For the y-coordinate: (4)^3 = 4^3 = 64

Thus, the coordinates of the transformed point would be (-27, 64).

3) To determine which of the given transformations could map the point (1,2) onto the point (3,6), we can substitute the x-coordinate and y-coordinate of point (1,2) into each transformation and check if they match the coordinates of (3,6).

a) T (to the 2,3 power):
For the x-coordinate: (1)^2 = 1
For the y-coordinate: (2)^3 = 8
Since (1,8) does not match (3,6), T (to the 2,3 power) does not map (1,2) to (3,6).

b) T (to the 3,2 power):
For the x-coordinate: (1)^3 = 1
For the y-coordinate: (2)^2 = 4
Again, (1,4) does not match (3,6), so T (to the 3,2 power) does not map (1,2) to (3,6).

c) D (to the 3rd power):
For the x-coordinate: (-1)^3 = -1
For the y-coordinate: (2)^3 = 8
Since (-1,8) does not match (3,6), D (to the 3rd power) does not map (1,2) to (3,6).

d) r (to the y=x power):
For the x-coordinate: (2)^1 = 2
For the y-coordinate: (1)^1 = 1
As (2,1) does not match (3,6), r (to the y=x power) does not map (1,2) to (3,6).

Therefore, none of the transformations listed can map (1,2) onto (3,6).

4) a) The equation for circle O with a center at the origin and a radius of 5 is:
x^2 + y^2 = 5^2
x^2 + y^2 = 25

b) After the translation of T (to the 2,-3 power), the center of circle O' would be (2, -3). Since translation does not affect the radius, the radius of O' would still be 5.

c) To write the equation for circle O', we can use the formula:
(x - h)^2 + (y - k)^2 = r^2
where (h, k) represents the center of the circle.

Using the information from part b), the equation for circle O' would be:
(x - 2)^2 + (y + 3)^2 = 5^2
(x - 2)^2 + (y + 3)^2 = 25