Question

activity 4

1. draw the image of region a after each of the following translations:

a) a is translated 2 units right and 3 units up to b
b) a is translated (5; 0) to c
c) a is translated (-4; -5) to d

Explanation:

Part (a)

To translate a figure 2 units right and 3 units up, we apply the translation rule \((x,y)\to(x + 2,y + 3)\) to each vertex of region \(A\).

1. Identify the vertices of region \(A\) from the graph. Let's assume the vertices of \(A\) are \((1,2)\), \((2,1)\), \((3,-1)\), and \((4,3)\) (by looking at the grid).
2. Apply the translation to each vertex:

• For \((1,2)\): \(x=1 + 2=3\), \(y = 2+3 = 5\), so the new vertex is \((3,5)\).
• For \((2,1)\): \(x=2 + 2=4\), \(y = 1+3 = 4\), so the new vertex is \((4,4)\).
• For \((3,-1)\): \(x=3 + 2=5\), \(y=-1 + 3=2\), so the new vertex is \((5,2)\).
• For \((4,3)\): \(x=4 + 2=6\), \(y = 3+3 = 6\), so the new vertex is \((6,6)\).

1. Plot these new vertices \((3,5)\), \((4,4)\), \((5,2)\), \((6,6)\) and connect them to get the image \(B\).

Part (b)

The translation vector \((5,0)\) means we move each vertex 5 units to the right (since the \(y\)-component is 0, there is no vertical movement). The translation rule is \((x,y)\to(x + 5,y)\).

1. Take the vertices of \(A\) \((1,2)\), \((2,1)\), \((3,-1)\), \((4,3)\).
2. Apply the translation:

• For \((1,2)\): \(x=1 + 5=6\), \(y = 2\), so the new vertex is \((6,2)\).
• For \((2,1)\): \(x=2 + 5=7\), \(y = 1\), so the new vertex is \((7,1)\).
• For \((3,-1)\): \(x=3 + 5=8\), \(y=-1\), so the new vertex is \((8,-1)\).
• For \((4,3)\): \(x=4 + 5=9\), \(y = 3\), so the new vertex is \((9,3)\).

1. Plot these new vertices \((6,2)\), \((7,1)\), \((8,-1)\), \((9,3)\) and connect them to get the image \(C\).

Part (c)

The translation vector \((- 4,-5)\) means we move each vertex 4 units to the left (negative \(x\)-direction) and 5 units down (negative \(y\)-direction). The translation rule is \((x,y)\to(x-4,y - 5)\).

1. Take the vertices of \(A\) \((1,2)\), \((2,1)\), \((3,-1)\), \((4,3)\).
2. Apply the translation:

• For \((1,2)\): \(x=1-4=-3\), \(y = 2-5=-3\), so the new vertex is \((-3,-3)\).
• For \((2,1)\): \(x=2-4=-2\), \(y = 1-5=-4\), so the new vertex is \((-2,-4)\).
• For \((3,-1)\): \(x=3-4=-1\), \(y=-1-5=-6\), so the new vertex is \((-1,-6)\).
• For \((4,3)\): \(x=4-4 = 0\), \(y = 3-5=-2\), so the new vertex is \((0,-2)\).

1. Plot these new vertices \((-3,-3)\), \((-2,-4)\), \((-1,-6)\), \((0,-2)\) and connect them to get the image \(D\).

(Note: Since the problem is about drawing the images, the above steps describe how to find the vertices of the translated figures. The actual drawing involves plotting these vertices on the given coordinate grid and connecting them to form the polygons \(B\), \(C\), and \(D\) corresponding to the translations of \(A\).)

Answer:

The translation vector \((- 4,-5)\) means we move each vertex 4 units to the left (negative \(x\)-direction) and 5 units down (negative \(y\)-direction). The translation rule is \((x,y)\to(x-4,y - 5)\).

1. Take the vertices of \(A\) \((1,2)\), \((2,1)\), \((3,-1)\), \((4,3)\).
2. Apply the translation:

• For \((1,2)\): \(x=1-4=-3\), \(y = 2-5=-3\), so the new vertex is \((-3,-3)\).
• For \((2,1)\): \(x=2-4=-2\), \(y = 1-5=-4\), so the new vertex is \((-2,-4)\).
• For \((3,-1)\): \(x=3-4=-1\), \(y=-1-5=-6\), so the new vertex is \((-1,-6)\).
• For \((4,3)\): \(x=4-4 = 0\), \(y = 3-5=-2\), so the new vertex is \((0,-2)\).

1. Plot these new vertices \((-3,-3)\), \((-2,-4)\), \((-1,-6)\), \((0,-2)\) and connect them to get the image \(D\).

(Note: Since the problem is about drawing the images, the above steps describe how to find the vertices of the translated figures. The actual drawing involves plotting these vertices on the given coordinate grid and connecting them to form the polygons \(B\), \(C\), and \(D\) corresponding to the translations of \(A\).)