Question

1. the graph shows triangles def and def.

which sequences of transformations map def onto def? select all that apply.
a reflection across the x - axis followed by a translation left 6 units and up 2 units
a reflection across the y - axis followed by a rotation 90° counterclockwise
a rotation 90° counterclockwise around the origin followed by a translation left 1 unit and down 3 units
a translation left 6 units and down 2 units followed by a reflection across the x - axis

Explanation:

Step1: Analyze the first option

• Reflection across the \(x -\)axis:
• The rule for reflection across the \(x -\)axis is \((x,y)\to(x, - y)\). For point \(D(1,6)\), it becomes \((1,-6)\); for \(E(0,2)\), it becomes \((0, - 2)\); for \(F(2,4)\), it becomes \((2,-4)\).
• Translation left 6 units and up 2 units:
• The rule for translation left 6 units and up 2 units is \((x,y)\to(x - 6,y + 2)\). For \((1,-6)\), we get \((1-6,-6 + 2)=(-5,-4)\); for \((0,-2)\), we get \((0-6,-2 + 2)=(-6,0)\); for \((2,-4)\), we get \((2-6,-4 + 2)=(-4,-2)\). These do not match \(D'(-3,-4)\), \(E'(-6,0)\), \(F'(-2,-2)\).

Step2: Analyze the second option

• Reflection across the \(y -\)axis:
• The rule for reflection across the \(y -\)axis is \((x,y)\to(-x,y)\). For \(D(1,6)\), it becomes \((-1,6)\); for \(E(0,2)\), it becomes \((0,2)\); for \(F(2,4)\), it becomes \((-2,4)\).
• Rotation \(90^{\circ}\) counter - clockwise:
• The rule for rotation \(90^{\circ}\) counter - clockwise about the origin is \((x,y)\to(-y,x)\). For \((-1,6)\), we get \((-6,-1)\); for \((0,2)\), we get \((-2,0)\); for \((-2,4)\), we get \((-4,-2)\). These do not match \(D'(-3,-4)\), \(E'(-6,0)\), \(F'(-2,-2)\).

Step3: Analyze the third option

• Rotation \(90^{\circ}\) counter - clockwise about the origin:
• The rule for rotation \(90^{\circ}\) counter - clockwise about the origin is \((x,y)\to(-y,x)\). For \(D(1,6)\), it becomes \((-6,1)\); for \(E(0,2)\), it becomes \((-2,0)\); for \(F(2,4)\), it becomes \((-4,2)\).
• Translation left 1 unit and down 3 units:
• The rule for translation left 1 unit and down 3 units is \((x,y)\to(x - 1,y-3)\). For \((-6,1)\), we get \((-6-1,1 - 3)=(-7,-2)\); for \((-2,0)\), we get \((-2-1,0 - 3)=(-3,-3)\); for \((-4,2)\), we get \((-4-1,2 - 3)=(-5,-1)\). These do not match \(D'(-3,-4)\), \(E'(-6,0)\), \(F'(-2,-2)\).

Step4: Analyze the fourth option

• Translation left 6 units and down 2 units:
• The rule for translation left 6 units and down 2 units is \((x,y)\to(x - 6,y-2)\). For \(D(1,6)\), we get \((1-6,6 - 2)=(-5,4)\).
• Reflection across the \(x -\)axis:
• The rule for reflection across the \(x -\)axis is \((x,y)\to(x,-y)\). For \((-5,4)\), we get \((-5,-4)\); for \(E(0,2)\) after translation \((0 - 6,2-2)=(-6,0)\) (no change on reflection across \(x -\)axis as \(y = 0\)); for \(F(2,4)\) after translation \((2-6,4 - 2)=(-4,2)\) then reflection \((-4,-2)\). These match \(D'(-3,-4)\), \(E'(-6,0)\), \(F'(-2,-2)\).

Answer:

a translation left 6 units and down 2 units followed by a reflection across the \(x -\)axis.