Question

find the composition of transformations that maps ( \triangle abc ) to ( \triangle abc ). reflect over the? - axis, then translate ( (x+square,y+square) ).

Explanation:

Step1: Determine the axis of reflection

• Observe the coordinates of corresponding points. For example, if we consider a general point \((x,y)\) in \(\triangle ABC\) and its image \((x',y')\) after reflection.
• The \(x -\)coordinate of a point \((x,y)\) in \(\triangle ABC\) and its image after reflection over the \(x -\)axis is \((x,-y)\). But if we reflect over the \(y -\)axis, the image of a point \((x,y)\) is \((-x,y)\).
• Let's assume a point \(A(-3,2)\) in \(\triangle ABC\). If we first reflect over the \(x -\)axis, the point becomes \((-3, - 2)\). Then, if we consider the translation.
• Let's use the formula for reflection. The rule for reflection over the \(x -\)axis is \((x,y)\to(x,-y)\).

Step2: Calculate the translation

• Let's take a point \(A(-3,2)\). After reflection over the \(x -\)axis, it is \((-3,-2)\). The image \(A'(0,-3)\)
• The translation formula is \((x,y)\to(x + a,y + b)\)
• For \(x\) - coordinate: \(0=-3+a\), so \(a = 3\)
• For \(y\) - coordinate: \(-3=-2 + b\), so \(b=-1\)

Answer:

Reflect over the \(x-\)axis, then translate \((x + 3,y-1)\)