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

Take point \(A(0, - 2)\) and its image \(A'(-3,-1)\).
For the \(x\) - coordinate: \(x\) changes from \(0\) to \(-3\), so \(x=x_1-3\).
For the \(y\) - coordinate: \(y\) changes from \(-2\) to \(-1\), so \(y=y_1 + 1\).
The translation rule is \((x,y)\to(x - 3,y + 1)\), which is equivalent to \(T_{-3,1}(x,y)\) in translation notation.

Step2: Check each option

• Option 1: The rule \(T_{-3,1}(x,y)\) is correct as we found from the point - to - point translation.
• Option 2: The rule \(T_{-1,3}(x,y)\) is incorrect.
• Option 3: The rule \((x,y)\to(x + 1,y-3)\) is incorrect.
• Option 4: The rule \((x,y)\to(x - 3,y + 1)\) is correct.
• Option 5: Trapezoid \(ABCD\) has been translated \(3\) units to the left (not right) and \(1\) unit up. So this option is incorrect.

Answer:

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