Question

graph the image of \\( \triangle lmn \\) after a translation 3 units up and 7 units right and a reflection over the line \\( y = - 2 \\).

Explanation:

Step1: Find coordinates of \( \triangle LMN \)

Assume \( L(-5,-1) \), \( M(-2,-2) \), \( N(-3,-3) \)

Step2: Apply translation

Translation rule: \( (x,y)\to(x + 7,y+3) \)

• For \( L(-5,-1) \): \( x=-5 + 7=2 \), \( y=-1+3 = 2\), so \( L'(2,2) \)
• For \( M(-2,-2) \): \( x=-2 + 7=5 \), \( y=-2+3 = 1\), so \( M'(5,1) \)
• For \( N(-3,-3) \): \( x=-3 + 7=4 \), \( y=-3+3 = 0\), so \( N'(4,0) \)

Step3: Apply reflection over \( y=-2 \)

Reflection rule over \( y = k\): \( (x,y)\to(x,2k - y) \), here \( k=-2 \), so \( (x,y)\to(x,-4 - y) \)

• For \( L'(2,2) \): \( x = 2\), \( y=-4-2=-6\), so \( L''(2,-6) \)
• For \( M'(5,1) \): \( x = 5\), \( y=-4 - 1=-5\), so \( M''(5,-5) \)
• For \( N'(4,0) \): \( x = 4\), \( y=-4-0=-4\), so \( N''(4,-4) \)

Step4: Graph the new triangle

Plot the points \( L''(2,-6) \), \( M''(5,-5) \), \( N''(4,-4) \) and connect them to form \( \triangle L''M''N'' \)

Answer:

Graph the points \( L''(2,-6) \), \( M''(5,-5) \), \( N''(4,-4) \) and connect them.