Question

exercises list the ordered pair of the points given. perform the translation using only arithmetic. then, list the ordered pair of the translated points. 1. $t_{(-1,1)}$ e (__, ) → e (, ) b (, ) → b (, ) 2. $t_{(3,2)}$ x (, ) → x (, ) y (, ) → y (, ) 3. $t_{(-3,4)}$ k (, ) → k (, ) h (, ) → h (, ) 4. $t_{(-5,1)}$ u (, ) → u (, ) p (, ) → p (, ) 5. $t_{(-10,-3)}$ b (, ) → b (, ) x (, ) → x (, ) 6. $t_{(-3,3)}$ k (, ) → k (, ) p (, ) → p (, __)

Explanation:

1. Assume we can read the coordinates of the points from the graph (not shown here with actual values). For a translation \(T_{(a,b)}\), if a point has coordinates \((x,y)\), the translated - point \((x',y')\) has coordinates \((x + a,y + b)\).

• For \(T_{(-1,1)}\):
• Let the coordinates of point \(E\) be \((x_E,y_E)\). Then the coordinates of \(E'\) are \((x_E-1,y_E + 1)\).
• Let the coordinates of point \(B\) be \((x_B,y_B)\). Then the coordinates of \(B'\) are \((x_B-1,y_B + 1)\).
• For \(T_{(3,2)}\):
• Let the coordinates of point \(X\) be \((x_X,y_X)\). Then the coordinates of \(X'\) are \((x_X + 3,y_X+2)\).
• Let the coordinates of point \(Y\) be \((x_Y,y_Y)\). Then the coordinates of \(Y'\) are \((x_Y + 3,y_Y+2)\).
• For \(T_{(-3,4)}\):
• Let the coordinates of point \(K\) be \((x_K,y_K)\). Then the coordinates of \(K'\) are \((x_K-3,y_K + 4)\).
• Let the coordinates of point \(H\) be \((x_H,y_H)\). Then the coordinates of \(H'\) are \((x_H-3,y_H + 4)\).
• For \(T_{(-5,1)}\):
• Let the coordinates of point \(U\) be \((x_U,y_U)\). Then the coordinates of \(U'\) are \((x_U-5,y_U + 1)\).
• Let the coordinates of point \(P\) be \((x_P,y_P)\). Then the coordinates of \(P'\) are \((x_P-5,y_P + 1)\).
• For \(T_{(-10,-3)}\):
• Let the coordinates of point \(B\) be \((x_B,y_B)\). Then the coordinates of \(B'\) are \((x_B-10,y_B-3)\).
• Let the coordinates of point \(X\) be \((x_X,y_X)\). Then the coordinates of \(X'\) are \((x_X-10,y_X-3)\).
• For \(T_{(-3,3)}\):
• Let the coordinates of point \(K\) be \((x_K,y_K)\). Then the coordinates of \(K'\) are \((x_K-3,y_K + 3)\).
• Let the coordinates of point \(P\) be \((x_P,y_P)\). Then the coordinates of \(P'\) are \((x_P-3,y_P + 3)\).

Since we don't have the actual coordinates of the points from the graph, we can't give the specific numerical - value answers. But the general rule for translation of a point \((x,y)\) by \(T_{(a,b)}\) is \((x + a,y + b)\).

If we assume some sample coordinates for illustration:

• Suppose \(E=(2,3)\) and \(T_{(-1,1)}\):
• # Explanation:
• ## Step 1: Apply translation rule for \(x\) - coordinate
• \(x_{E'}=x_E+a\), where \(x_E = 2\) and \(a=-1\), so \(x_{E'}=2+( - 1)=1\).
• ## Step 2: Apply translation rule for \(y\) - coordinate
• \(y_{E'}=y_E + b\), where \(y_E = 3\) and \(b = 1\), so \(y_{E'}=3 + 1=4\). So \(E'=(1,4)\).
• Suppose \(X=(4,5)\) and \(T_{(3,2)}\):
• ## Step 1: Apply translation rule for \(x\) - coordinate
• \(x_{X'}=x_X+a\), where \(x_X = 4\) and \(a = 3\), so \(x_{X'}=4+3 = 7\).
• ## Step 2: Apply translation rule for \(y\) - coordinate
• \(y_{X'}=y_X + b\), where \(y_X = 5\) and \(b = 2\), so \(y_{X'}=5 + 2=7\). So \(X'=(7,7)\).

In general, for a point \((x,y)\) and a translation \(T_{(a,b)}\), the new - point \((x',y')\) has coordinates \(x'=x + a\) and \(y'=y + b\).

If we had the actual coordinates of the points from the graph, we would substitute the values of \(x\) and \(y\) (the original coordinates) and \(a\) and \(b\) (the translation values) into the above formulas to get the exact coordinates of the translated points.

Answer:

1. Assume we can read the coordinates of the points from the graph (not shown here with actual values). For a translation \(T_{(a,b)}\), if a point has coordinates \((x,y)\), the translated - point \((x',y')\) has coordinates \((x + a,y + b)\).

• For \(T_{(-1,1)}\):
• Let the coordinates of point \(E\) be \((x_E,y_E)\). Then the coordinates of \(E'\) are \((x_E-1,y_E + 1)\).
• Let the coordinates of point \(B\) be \((x_B,y_B)\). Then the coordinates of \(B'\) are \((x_B-1,y_B + 1)\).
• For \(T_{(3,2)}\):
• Let the coordinates of point \(X\) be \((x_X,y_X)\). Then the coordinates of \(X'\) are \((x_X + 3,y_X+2)\).
• Let the coordinates of point \(Y\) be \((x_Y,y_Y)\). Then the coordinates of \(Y'\) are \((x_Y + 3,y_Y+2)\).
• For \(T_{(-3,4)}\):
• Let the coordinates of point \(K\) be \((x_K,y_K)\). Then the coordinates of \(K'\) are \((x_K-3,y_K + 4)\).
• Let the coordinates of point \(H\) be \((x_H,y_H)\). Then the coordinates of \(H'\) are \((x_H-3,y_H + 4)\).
• For \(T_{(-5,1)}\):
• Let the coordinates of point \(U\) be \((x_U,y_U)\). Then the coordinates of \(U'\) are \((x_U-5,y_U + 1)\).
• Let the coordinates of point \(P\) be \((x_P,y_P)\). Then the coordinates of \(P'\) are \((x_P-5,y_P + 1)\).
• For \(T_{(-10,-3)}\):
• Let the coordinates of point \(B\) be \((x_B,y_B)\). Then the coordinates of \(B'\) are \((x_B-10,y_B-3)\).
• Let the coordinates of point \(X\) be \((x_X,y_X)\). Then the coordinates of \(X'\) are \((x_X-10,y_X-3)\).
• For \(T_{(-3,3)}\):
• Let the coordinates of point \(K\) be \((x_K,y_K)\). Then the coordinates of \(K'\) are \((x_K-3,y_K + 3)\).
• Let the coordinates of point \(P\) be \((x_P,y_P)\). Then the coordinates of \(P'\) are \((x_P-3,y_P + 3)\).

Since we don't have the actual coordinates of the points from the graph, we can't give the specific numerical - value answers. But the general rule for translation of a point \((x,y)\) by \(T_{(a,b)}\) is \((x + a,y + b)\).

If we assume some sample coordinates for illustration:

• Suppose \(E=(2,3)\) and \(T_{(-1,1)}\):
• # Explanation:
• ## Step 1: Apply translation rule for \(x\) - coordinate
• \(x_{E'}=x_E+a\), where \(x_E = 2\) and \(a=-1\), so \(x_{E'}=2+( - 1)=1\).
• ## Step 2: Apply translation rule for \(y\) - coordinate
• \(y_{E'}=y_E + b\), where \(y_E = 3\) and \(b = 1\), so \(y_{E'}=3 + 1=4\). So \(E'=(1,4)\).
• Suppose \(X=(4,5)\) and \(T_{(3,2)}\):
• ## Step 1: Apply translation rule for \(x\) - coordinate
• \(x_{X'}=x_X+a\), where \(x_X = 4\) and \(a = 3\), so \(x_{X'}=4+3 = 7\).
• ## Step 2: Apply translation rule for \(y\) - coordinate
• \(y_{X'}=y_X + b\), where \(y_X = 5\) and \(b = 2\), so \(y_{X'}=5 + 2=7\). So \(X'=(7,7)\).

In general, for a point \((x,y)\) and a translation \(T_{(a,b)}\), the new - point \((x',y')\) has coordinates \(x'=x + a\) and \(y'=y + b\).

If we had the actual coordinates of the points from the graph, we would substitute the values of \(x\) and \(y\) (the original coordinates) and \(a\) and \(b\) (the translation values) into the above formulas to get the exact coordinates of the translated points.