Question

which statements are true about trapezoid abcd and its translated image, abcd? select two options. the rule for the translation can be written as ( t_{-3,1}(x,y) ). the rule for the translation can be written as ( t_{-1,3}(x,y) ). the rule for the translation can be written as ( (x,y)\to(x + 1,y - 3) ). the rule for the translation can be written as ( (x,y)\to(x - 3,y + 1) ). trapezoid abcd has been translated 3 units to the right and 1 unit up.

Explanation:

Step1: Analyze the translation of a point

Let's take point \(A(0, - 2)\) and its image \(A'(-3,-1)\).
For the \(x\) - coordinate: \(x\) changes from \(0\) to \(-3\), so \(\Delta x=-3 - 0=-3\).
For the \(y\) - coordinate: \(y\) changes from \(-2\) to \(-1\), so \(\Delta y=-1-(-2) = 1\).
The translation rule is \((x,y)\to(x - 3,y + 1)\) which can also be written as \(T_{-3,1}(x,y)\) (where \(T_{a,b}(x,y)=(x + a,y + b)\))

Step2: Check each option

• Option 1: The rule \(T_{-3,1}(x,y)=(x-3,y + 1)\) is correct as we found from the point - to - point translation.
• Option 2: For \(T_{-1,3}(x,y)=(x-1,y + 3)\). If \(x = 0,y=-2\), then \((x-1,y + 3)=(-1,1)

eq A'(-3,-1)\), so this option is wrong.

• Option 3: For \((x,y)\to(x + 1,y-3)\). If \(x = 0,y=-2\), then \((x + 1,y-3)=(1,-5)

eq A'(-3,-1)\), so this option is wrong.

• Option 4: The rule \((x,y)\to(x - 3,y + 1)\) is correct as we found from the point - to - point translation.
• Option 5: Trapezoid \(ABCD\) has been translated \(3\) units to the left (since \(x\) - coordinate decreases by \(3\)) and \(1\) unit up (since \(y\) - coordinate increases by \(1\)), so this option is wrong.

Answer:

The rule for the translation can be written as \(T_{-3,1}(x,y)\) and The rule for the translation can be written as \((x,y)\to(x - 3,y + 1)\)