Can someone explain to me how this question has the answer like this

Points:(2,3)

y=-2f[2(x+5)] - 4

The answers are (2, 3) ( 2, -3) -> (2, -6)-> (1, -6) -> (-4, -10)

I know it has to do with translations like 1/k etc.. but im a bit confused ):

To understand how the points were obtained, let's analyze the given equation step by step.

The equation y = -2f[2(x+5)] - 4 consists of three main parts: the inner function, the transformation, and the constant term.

1. Inner Function: (x+5)
The inner function (x+5) implies a horizontal translation of 5 units to the left. This means that any x-coordinate value in the equation will be shifted 5 units to the left.

2. Transformation: -2f[2(x+5)]
The notation f[ ] indicates a vertical scaling factor. In this case, the vertical scale factor is 2, which means that the y-coordinate of any point in the equation will be multiplied by 2.

3. Constant Term: -4
The constant term of -4 represents a vertical translation downward by 4 units. This means that the y-coordinate of any point in the equation will be shifted 4 units downward.

Now, let's apply these transformations step by step with the given point (2,3) and see how the answers are obtained:

1. Starting with the point (2,3):
The inner function (x+5) will shift x=2 five units to the left:
(2+5, 3) = (7,3)

2. Apply the vertical scaling factor of 2:
The y-coordinate of (7,3) will be multiplied by 2:
(7, 3*2) = (7, 6)

3. Apply the vertical translation downward by 4 units:
The y-coordinate of (7,6) will be shifted 4 units down:
(7, 6-4) = (7, 2)

Therefore, the point (7,2) is the result after applying all the transformations to the original point (2,3).

To obtain the other answers, you need to follow the same transformation steps with the new point obtained:

4. The point (7,2) becomes (2,2) as the x-coordinate is shifted back 5 units.

5. The vertical scaling is applied:
(2, 2*2) = (2, 4)

6. The vertical translation downwards is applied:
(2, 4-4) = (2, 0)

Thus, the point (2,0) is obtained.

By continuing these steps, you get:
(2,3) -> (7,2) -> (2,0) -> (1,0) -> (-4,-2) -> (-4,-10)

So, the answers you mentioned (2, 3), (2, -3), (2, -6), (1, -6), (-4, -10) are the result of applying the transformations to the original point (2,3).