Question

**#3.) what is a sequence of transformations that would map ab onto ab? think: was the figure slid, flipped, or turned?
e) ( r_{y - axis} circ t_{-9,-7} )
f) ( r_{-90} circ r_{y - axis} )
g) ( r_{180} circ t_{-9,-7} )
h) ( r_{-90} circ r_{x - axis} )

Explanation:

Step1: Analyze each option

• Option e: \(T_{-9,-7}\) is a translation (slide) by \((- 9,-7)\) and \(r_{y - axis}\) is a reflection over the \(y\) - axis.
• Option f: \(r_{y - axis}\) reflects over the \(y\) - axis and \(R_{-90}\) rotates \(90^{\circ}\) clockwise.
• Option g: \(T_{-9,-7}\) is a translation and \(R_{180}\) is a \(180^{\circ}\) rotation.
• Option h: \(r_{x - axis}\) reflects over the \(x\) - axis and \(R_{-90}\) rotates \(90^{\circ}\) clockwise.

Step2: Use coordinate analysis

Let \(A=(x_1,y_1)\) and \(B=(x_2,y_2)\).

• Assume \(A = (-4,6)\) and \(B=(-4,0)\) (from the grid).
• For \(r_{x - axis}\): \((x,y)\to(x, - y)\). So \(A=(-4,6)\to A'=(-4,-6)\) and \(B = (-4,0)\to B'=(-4,0)\)
• For \(R_{-90}\): The rotation rule \((x,y)\to(y,-x)\). If we have a point \((x,y)\) after reflection over \(x\) - axis \((x,-y)\), then after \(R_{-90}\) rotation \((-y,-x)\)
• If we first reflect \(AB\) over the \(x\) - axis: \(A(-4,6)\to A_1(-4, - 6)\), \(B(-4,0)\to B_1(-4,0)\)
• Then rotate \(R_{-90}\): \(A_1(-4,-6)\to A''( - 6,4)\) (incorrect).
• Let's check option f:
• Reflect \(A(-4,6)\) over \(y\) - axis: \(A'=(4,6)\), \(B(-4,0)\to B'=(4,0)\)
• Rotate \(R_{-90}\): The rule for \(R_{-90}\) is \((x,y)\to(y,-x)\). So \(A'(4,6)\to A''(6,-4)\) (incorrect)
• Let's check option e:
• Translate \(A(-4,6)\) by \(T_{-9,-7}\): \(A_1=(-4 - 9,6-7)=(-13,-1)\), then reflect over \(y\) - axis: \(A''=(13,-1)\) (incorrect)
• Let's check option h:
• Reflect \(A(-4,6)\) over \(x\) - axis: \(A_1=(-4,-6)\)
• Rotate \(R_{-90}\): Using the rule \((x,y)\to(y,-x)\), for \(A_1(-4,-6)\), we get \(A''=(-6,4)\) (incorrect)

Answer:

h) \(R_{-90}\circ r_{x - axis}\)