Question

the graph shows △abc and its image △abc. describe a sequence of transformations that would map △abc on to △abc. reflection over x - axis, then translated according to the rule (x, y) → (x - 4, y - 2). reflection over the y - axis, then translated according to the rule (x, y) → (x - 2, y - 4). translated according to the rule (x, y) → (x - 2, y - 4), then reflected over the x - axis. translated according to the rule (x, y) → (x - 4, y - 2), then reflected over the y - axis.

Explanation:

Step1: Analyze Reflection over y - axis

First, consider reflecting \(\triangle ABC\) over the \(y\) - axis. The rule for reflection over the \(y\) - axis is \((x,y)\to(-x,y)\). Let's take a point from \(\triangle ABC\), say \(C(-1,4)\) (wait, actually looking at the graph, \(C\) is at \((-2,4)\)? Wait, no, from the grid, let's re - identify the coordinates. Let's find the coordinates of \(\triangle ABC\): Let's assume \(A(-8,3)\), \(B(-5,1)\), \(C(-2,4)\) (by looking at the grid). After reflecting over the \(y\) - axis, \(A\) becomes \((8,3)\), \(B\) becomes \((5,1)\), \(C\) becomes \((2,4)\).

Step2: Analyze Translation \((x,y)\to(x - 2,y - 4)\)

Now, apply the translation \((x,y)\to(x - 2,y - 4)\) to the reflected points. For the reflected \(A(8,3)\): \(x=8 - 2=6\), \(y = 3-4=-1\), which is \(A''(6,-1)\) (matches the graph). For reflected \(B(5,1)\): \(x = 5-2 = 3\), \(y=1 - 4=-3\), which is \(B''(3,-3)\) (matches the graph). For reflected \(C(2,4)\): \(x=2 - 2=0\), \(y = 4-4 = 0\), which is \(C''(0,0)\) (matches the graph).

Let's check the other options:

• Option 1: Reflection over \(x\) - axis then translation \((x,y)\to(x - 4,y - 2)\). Reflecting \(A(-8,3)\) over \(x\) - axis gives \((-8,-3)\), translating gives \((-8 - 4,-3-2)=(-12,-5)

eq A''(6,-1)\).

• Option 3: Translation \((x,y)\to(x - 2,y - 4)\) then reflection over \(x\) - axis. Translating \(A(-8,3)\) gives \((-8 - 2,3 - 4)=(-10,-1)\), reflecting over \(x\) - axis gives \((-10,1)

eq A''(6,-1)\).

• Option 4: Translation \((x,y)\to(x - 4,y - 2)\) then reflection over \(y\) - axis. Translating \(A(-8,3)\) gives \((-8 - 4,3 - 2)=(-12,1)\), reflecting over \(y\) - axis gives \((12,1)

eq A''(6,-1)\).

Answer:

Reflection over the \(y\) - axis, then translated according to the rule \((x,y)\to(x - 2,y - 4)\) (the second option in the given choices, which is the one with the blue circle next to it: "Reflection over the \(y\) - axis, then translated according to the rule \((x,y)\to(x - 2,y - 4)\)")