Question

1. a translation has the rule (x,y)→(x + 3,y - 3). find the image of x(2,5). find the preimage of y(4,4). 4. a triangle has vertices a(7,11), b(-5,1), and c(4,-2). after a translation, it is known that a(8,10). write a rule for the translation: find the coordinates for b and c:

Explanation:

3.

Find the image of \(X(2,5)\)

Step1: Apply \(x\) - coordinate rule

For the \(x\) - coordinate of the image, use the rule \(x
ightarrow x + 3\). Given \(x = 2\), then \(x'=2 + 3=5\).

Step2: Apply \(y\) - coordinate rule

For the \(y\) - coordinate of the image, use the rule \(y
ightarrow y - 3\). Given \(y = 5\), then \(y'=5 - 3 = 2\).

Step1: Set up equations for pre - image

Let the pre - image be \((x,y)\). The translation rule is \((x,y)
ightarrow(x + 3,y - 3)\). So we have the equations \(x+3 = 4\) and \(y - 3=4\).

Step2: Solve for \(x\)

Solve \(x+3 = 4\) for \(x\). Subtract 3 from both sides: \(x=4 - 3=1\).

Step3: Solve for \(y\)

Solve \(y - 3=4\) for \(y\). Add 3 to both sides: \(y=4 + 3 = 7\).

Step1: Find the change in \(x\) - coordinate

Given \(A(7,11)\) and \(A'(8,10)\). The change in the \(x\) - coordinate is \(\Delta x=8 - 7=1\).

Step2: Find the change in \(y\) - coordinate

The change in the \(y\) - coordinate is \(\Delta y=10 - 11=-1\).
The translation rule is \((x,y)
ightarrow(x + 1,y - 1)\).

Answer:

\((5,2)\)

Find the pre - image of \(Y'(4,4)\)