how is the transformation y=f(x+3)^2+2 affected. f(x) has the vector of (2, -6) so it would move left 3 units and up 2 so the new vector would be (-1,-4)... is that correct
Yes, your reasoning is correct. Let's break it down step by step.
The given transformation equation is y = f(x+3)^2 + 2, where f(x) represents the function with the vector (2, -6).
1. Horizontal shift: The term (x+3) in the equation represents a horizontal shift of 3 units to the left. So, if you start with the point (x, y) and perform the shift, the new x-coordinate would be (x+3).
2. Finding f(x+3): Since f(x) has the vector (2, -6), we need to find f(x+3). To do this, we shift the input by 3 units to the left. So, we replace all instances of x in the vector with (x+3), giving (2, -6) -> (2+3, -6) -> (5, -6).
3. Squaring: The term (x+3)^2 represents squaring the x-coordinate obtained after shifting. In this case, (5)^2 = 25.
4. Vertical shift: Finally, adding 2 to the squared result gives 25 + 2 = 27.
Therefore, the new vector would be (x+3, f(x+3)^2 + 2) = (-1, 27), which corresponds to a horizontal shift of 3 units to the left and a vertical shift of 2 units upward.
So, your answer of (-1, -4) is not correct. The correct answer is (-1, 27).