Question

the blue figure is a translation image of the black figure. write a rule to describe the translation. the translation rule is t(x,y).

Explanation:

Step1: Determine the horizontal translation

Pick a point from the black figure (pre - image) and its corresponding point in the blue figure (image). Let's say we take a vertex of the black triangle. Suppose a vertex of the black triangle has an \(x\) - coordinate. If we move from the black figure to the blue figure, we count the number of units in the \(x\) - direction.
If we assume a point \((x_1,y_1)\) in the black figure and its corresponding point \((x_2,y_2)\) in the blue figure. By observing the grid, if we take a point in the black figure (say \(x = 1\)) and its corresponding point in the blue figure (\(x=-2\)), the change in the \(x\) - direction is \(\Delta x=-2 - 1=-3\). In general, for a point \((x,y)\) in the black figure, the \(x\) - coordinate of the corresponding point in the blue figure is \(x-3\).

Step2: Determine the vertical translation

Now, for the \(y\) - coordinate. Using the same pair of corresponding points. If the \(y\) - coordinate of a point in the black figure is \(y_1\) and in the blue figure is \(y_2\). By observing the grid, if \(y_1=-1\) and \(y_2 = 2\), the change in the \(y\) - direction is \(\Delta y=2-(-1) = 3\). In general, for a point \((x,y)\) in the black figure, the \(y\) - coordinate of the corresponding point in the blue figure is \(y + 3\).

The translation rule \(T_{(a,b)}(x,y)=(x + a,y + b)\)

Answer:

\(T_{(-3,3)}(x,y)\)