Question

1. apply a translation left 3 and down 4 followed by a reflection over the x - axis to point (-4,1). what is the original point? (3,4) (4,-1) (-4,1) (-1,3)

Explanation:

Step1: Reverse the reflection

When reflecting over the \(x -\)axis, the rule is \((x,y)\to(x, - y)\). To reverse it, if the final \(y -\)coordinate is \(1\), then before reflection \(y=- 1\).

Step2: Reverse the translation

The translation rule is \((x,y)\to(x - 3,y - 4)\). To reverse it, we use \((x,y)\to(x + 3,y + 4)\). Given the final \(x=-4\), then \(x=-4 + 3=-1\); given the \(y\) - coordinate before reflection \(y = - 1\), then \(y=-1+4 = 3\). But wait, let's do it in formula.
Let the original point be \((x,y)\).
After translation: \((x-3,y - 4)\)
After reflection over \(x -\)axis: \((x-3,-(y - 4))=(x - 3,-y + 4)\)
We know \((x-3,-y + 4)=(-4,1)\)
Set up equations:
\(x-3=-4\), solve for \(x\): \(x=-4 + 3=-1\)
\(-y + 4=1\), solve for \(y\): \(-y=1 - 4=-3\), \(y = 3\)

Answer:

\((-1,3)\)