Question

find the coordinates of the vertices of the image of math for each transformation.

1. translation $(x,y)\to(x - 1,y + 2)$

$m$ ( type your answer... type your answer... ) and $a$ ( type your answer... type your answer... )

1. reflection across the $x$-axis

$m$ ( type your answer... type your answer... ) and $t$ ( type your answer... type your answer... )

1. counter-clockwise rotation 90 degrees about point $(0,0)$

$m$ ( type your answer... type your answer... ) and $h$ ( type your answer... type your answer... )

1. dilation with a scale factor of 4

$m$ ( type your answer... type your answer... ) $h$ ( type your answer... type your answer... )

1. apply the sequence with translation $(x,y)\to(x - 4,y)$ and reflection across the line $y=-2$

$m$ ( type your answer... type your answer... ) $m$ ( type your answer... type your answer... )

Explanation:

Step1: Find original coordinates

From the graph, \(M(-2,2)\), \(A(3,0)\), \(T(0,-4)\), \(H(-3,-2)\)

Step2: Solve for translation \((x,y)\to(x - 1,y+2)\)

For \(M\): \(x=-2-1=-3\), \(y = 2+2=4\), so \(M'(-3,4)\)
For \(A\): \(x=3-1=2\), \(y=0 + 2=2\), so \(A'(2,2)\)

Step3: Solve for reflection across the \(x\) - axis \((x,y)\to(x,-y)\)

For \(M\): \(x=-2\), \(y=-2\), so \(M'(-2,-2)\)
For \(T\): \(x = 0\), \(y = 4\), so \(T'(0,4)\)

Step4: Solve for counter - clockwise rotation \(90^{\circ}\) about \((0,0)\) \((x,y)\to(-y,x)\)

For \(M\): \(x=-2\), \(y=-2\), so \(M'(-2,-2)\)
For \(H\): \(x = 2\), \(y=-3\), so \(H'(2,-3)\)

Step5: Solve for dilation with scale factor \(4\) \((x,y)\to(4x,4y)\)

For \(M\): \(x=-2\times4=-8\), \(y=2\times4 = 8\), so \(M'(-8,8)\)
For \(H\): \(x=-3\times4=-12\), \(y=-2\times4=-8\), so \(H'(-12,-8)\)

Step6: Solve for the sequence: translation \((x,y)\to(x - 4,y)\) then reflection across \(y=-2\)

First, translation for \(M(-2,2)\): \(x=-2-4=-6\), \(y = 2\), so \(M_1(-6,2)\)
Then reflection across \(y=-2\): The distance from \(y = 2\) to \(y=-2\) is \(4\). So the new \(y\) - coordinate is \(-2-4=-6\), \(x=-6\), so \(M'(-6,-6)\)

Answer:

1. \(M'(-3,4)\), \(A'(2,2)\)
2. \(M'(-2,-2)\), \(T'(0,4)\)
3. \(M'(2,-2)\), \(H'(2,-3)\)
4. \(M'(-8,8)\), \(H'(-12,-8)\)
5. \(M'(-6,-6)\)