Question

which pair of transformations is equivalent to the transformation described by $(x,y)\to(x, - y + 1)$?
a a reflection across the $x$-axis followed by a translation of 1 unit up
b a translation of 1 unit up followed by a reflection across the $x$-axis
c a reflection across the $y$-axis followed by a translation of 1 unit up
d a translation of 1 unit up followed by a reflection across the $y$-axis

Explanation:

Step1: Analyze reflection across x - axis

The rule for reflection across the \(x\) - axis is \((x,y)\to(x, - y)\).

Step2: Analyze translation of 1 unit up

The rule for translation of 1 unit up is \((x,y)\to(x,y + 1)\).
If we first reflect \((x,y)\) across the \(x\) - axis to get \((x,-y)\), and then translate 1 unit up, we use the rule \((x,-y)\to(x,-y + 1)\) which is the given transformation \((x,y)\to(x,-y + 1)\).

Answer:

A. a reflection across the \(x\) - axis followed by a translation of 1 unit up