Question

figure wxyz is transformed using the rule $r_{y-axis} \circ t_{-4, 2}(x, y)$. point w of the pre-image is at (1, 6). what are the coordinates of point $w$ on the final image? (3, 8) (-3, -8) (5, -8) (-5, 8)

Explanation:

Step1: Apply Translation \( T_{-4,2} \)

The translation rule \( T_{a,b}(x,y)=(x + a,y + b) \). Here, \( a=-4 \), \( b = 2 \). For point \( W(1,6) \), after translation:
\( (1-4,6 + 2)=(-3,8) \)

Step2: Apply Reflection \( r_{y\text{-axis}} \)

The reflection over \( y \)-axis rule is \( r_{y\text{-axis}}(x,y)=(-x,y) \). Apply to \( (-3,8) \):
\( -(-3),8=(3,8) \)? Wait, no, wait. Wait, maybe I mixed the order? Wait, the composition is \( r_{y\text{-axis}}\circ T_{-4,2} \), so first translate, then reflect. Wait, no, function composition: \( (r_{y\text{-axis}}\circ T_{-4,2})(x,y)=r_{y\text{-axis}}(T_{-4,2}(x,y)) \). So first \( T_{-4,2}(1,6)=(-3,8) \), then reflect over \( y \)-axis: \( r_{y\text{-axis}}(-3,8)=(3,8) \)? But wait, maybe the translation is \( T_{-4,-2} \)? Wait, the original problem says \( T_{-4,2} \)? Wait, maybe a typo, but let's check the options. Wait, maybe the translation is \( T_{-4,-2} \). Let's re - calculate. If translation is \( T_{-4,-2} \): \( (1-4,6-2)=(-3,4) \)? No. Wait, maybe the reflection is over \( x \)-axis? Wait, the options have \( (-3,-8) \), \( (5,-8) \), \( (-5,8) \), \( (3,8) \). Let's re - examine.

Wait, maybe the translation is \( T_{-4,2} \) is wrong. Wait, the composition is \( r_{y\text{-axis}}\circ T_{-4,2} \). Let's do it correctly.

First, translation \( T_{-4,2} \): \( (x,y)\to(x - 4,y + 2) \). So \( (1,6)\to(1-4,6 + 2)=(-3,8) \).

Then reflection over \( y \)-axis: \( (x,y)\to(-x,y) \). So \( (-3,8)\to(3,8) \). But that's one of the options. Wait, but maybe the translation is \( T_{-4,-2} \)? Wait, if translation is \( T_{-4,-2} \): \( (1-4,6-2)=(-3,4) \), then reflection over \( x \)-axis: \( (x,y)\to(x,-y) \), so \( (-3,4)\to(-3,-4) \), not in options. Wait, maybe the reflection is over \( x \)-axis. Let's try reflection over \( x \)-axis first? No, composition is \( r_{y\text{-axis}}\circ T_{-4,2} \). Wait, maybe the original translation is \( T_{4,-2} \)? No. Wait, the first option is \( (3,8) \), let's check again.

Wait, point \( W(1,6) \). Translate by \( T_{-4,2} \): \( x = 1-4=-3 \), \( y = 6 + 2 = 8 \). Then reflect over \( y \)-axis: \( x=-(-3)=3 \), \( y = 8 \). So the result is \( (3,8) \).

Answer:

\( (3,8) \) (the first option: \( (3,8) \))