Question

writing a rule to describe a translation
(a) the arrows below show that the coordinates on the left are mapped to the coordinates on the right. fill in the blanks to give the coordinates after the translation.
original coordinates → final coordinates
$x(4,7)→x(\square,\square)$
$y(8,8)→y(\square,\square)$
$z(7,3)→z(\square,\square)$
(b) choose the general rule below that describes the translation mapping $\triangle xyz$ to $\triangle xyz$.
$\bigcirc (x,y)→(-8x,-5y)$ $\bigcirc (x,y)→(8x,5y)$
$\bigcirc (x,y)→(5x,8y)$ $\bigcirc (x,y)→(-5x,-8y)$
$\bigcirc (x,y)→(x + 5,y + 8)$ $\bigcirc (x,y)→(x - 8,y - 5)$
$\bigcirc (x,y)→(x + 8,y + 5)$ $\bigcirc (x,y)→(x - 5,y - 8)$

Explanation:

Step1: Find the coordinates of \(X'\)

From the graph, \(X(4,7)\) is translated to \(X'(- 4,-5)\).

Step2: Find the coordinates of \(Y'\)

From the graph, \(Y(8,8)\) is translated to \(Y'(0,-5)\).

Step3: Find the coordinates of \(Z'\)

From the graph, \(Z(7,3)\) is translated to \(Z'(-1,-5)\).

Step4: Determine the translation rule

For \(x\) - coordinate: \(4-8=-4\), \(8 - 8=0\), \(7-8=-1\) (subtract \(8\) from \(x\) - coordinate).
For \(y\) - coordinate: \(7-5 = 2\), \(8-5=3\), \(3-5=-2\) (subtract \(5\) from \(y\) - coordinate). The rule is \((x,y)\to(x - 8,y - 5)\)

Answer:

a) \(X'(-4,-5)\), \(Y'(0,-5)\), \(Z'(-1,-5)\)
b) \((x,y)\to(x - 8,y - 5)\)