Question

reflect the point $(-4,1)$ across the line $y = -2$, then translate it horizontally five units in the positive direction. what are the intermediate and the final coordinates, respectively? $(-4,-5),(1,-5)$ $(0,1),(5,1)$ $(-4,-1),(1,-1)$ $(0,1),(0,6)$

Explanation:

Step1: Reflect the point across the line \(y = - 2\)

The formula for reflecting a point \((x,y)\) across the line \(y = k\) is \((x,2k - y)\). Here \(x=-4,y = 1,k=-2\).
Substitute into the formula: \(x=-4\), \(2k - y=2\times(-2)-1=-4 - 1=-5\). So the intermediate point is \((-4,-5)\).

Step2: Translate the intermediate point horizontally

The formula for translating a point \((x,y)\) horizontally \(h\) units (positive \(h\) for right - ward) is \((x + h,y)\). Here \(x=-4,y=-5,h = 5\).
Substitute into the formula: \(x+h=-4 + 5=1\), \(y=-5\). So the final point is \((1,-5)\).

Answer:

A. \((-4,-5),(1,-5)\)