Question

1. smp.3 error analysis describe and correct the error in graphing the image of \\( \triangle a b c \\) after the translation \\( ( x, y ) \

ightarrow ( x - 3, y + 2 ) \\).

Explanation:

Step1: Understand the translation rule

The translation rule is \((x,y)\to(x - 3,y+2)\). This means for each point \((x,y)\) in \(\triangle ABC\), we subtract \(3\) from the \(x -\) coordinate (move \(3\) units to the left) and add \(2\) to the \(y -\) coordinate (move \(2\) units up).

Step2: Analyze the error in the given graph

Looking at the \(x -\) coordinate transformation: In the incorrect graph, it seems that the \(x -\) coordinate was increased (for example, if we assume a point \(A\) with original \(x -\) coordinate \(x_A\), in the wrong graph \(x_{A'}\) is greater than \(x_A\) instead of \(x_{A'}=x_A - 3\)). The \(y -\) coordinate transformation was also wrong. The correct \(y -\) coordinate should be \(y+2\) (up - shift), but in the given graph, the \(y -\) coordinate was decreased.

Step3: Correctly apply the translation rule

Let's assume \(A=(5,6)\), \(B=(3,3)\), \(C=(0,3)\).
For point \(A\):
Using the rule \((x,y)\to(x - 3,y + 2)\), we have \(x=5,y = 6\). Then \(x'=5-3 = 2\), \(y'=6 + 2=8\), so \(A'=(2,8)\).
For point \(B\):
\(x = 3,y=3\). Then \(x'=3-3=0\), \(y'=3 + 2 = 5\), so \(B'=(0,5)\).
For point \(C\):
\(x=0,y = 3\). Then \(x'=0-3=-3\), \(y'=3 + 2=5\), so \(C'=(-3,5)\).

Answer:

The error was that the \(x -\) coordinate was increased (instead of decreased by \(3\)) and the \(y -\) coordinate was decreased (instead of increased by \(2\)). The correct coordinates of the image of \(\triangle ABC\) after the translation \((x,y)\to(x - 3,y + 2)\): if \(A=(5,6)\), \(B=(3,3)\), \(C=(0,3)\), then \(A'=(2,8)\), \(B'=(0,5)\), \(C'=(-3,5)\)