Question

a triangle on a grid is reflected across the x - axis, then translated 3 units up. kai wants to describe the overall movement from the original to the final triangle.
question 1
if one vertex of the original triangle is at (2, - 4), where is this vertex after both transformations?
(-2,7)
(2,7)
(2,4)
(2, - 7)
question 2
kai wonders what would happen if she reversed the order of transformations: translate 3 units up first, then reflect across the x - axis. compared to her original result, the final triangle would be:
in the exact same position
6 units lower
reflected across the y - axis
6 units higher

Explanation:

Step1: Reflect across the x - axis

When a point \((x,y)\) is reflected across the \(x\) - axis, the rule is \((x,y)\to(x, - y)\).
For the point \((2,-4)\), after reflection across the \(x\) - axis, it becomes \((2,4)\).

Step2: Translate 3 units up

When a point \((x,y)\) is translated \(k\) units up, the rule is \((x,y)\to(x,y + k)\).
Here \(k = 3\), so for the point \((2,4)\), after translation 3 units up, it becomes \((2,4 + 3)=(2,7)\).

Step3: Analyze the second - part (reverse transformation)

Let the original point be \((x,y)\).
First translation 3 units up: \((x,y)\to(x,y + 3)\)
Then reflection across the \(x\) - axis: \((x,y+3)\to(x,-(y + 3))=(x,-y-3)\)
Original transformation:
First reflection across the \(x\) - axis: \((x,y)\to(x,-y)\)
Then translation 3 units up: \((x,-y)\to(x,-y + 3)\)
The difference between \((x,-y + 3)\) and \((x,-y-3)\) is \((x,-y + 3)-(x,-y-3)=(0,6)\) (in terms of vector subtraction). The second result is 6 units lower than the first result.

Answer:

Question 1: B. \((2,7)\)
Question 2: B. 6 units lower