Question

pentagon tuvwx is the image of pentagon tuvwx under a reflection followed by a translation right 4 units and down 3 units. write the rules for the reflection and translation. reflection: (x, y) ↦ (\boxed{ }, \boxed{ }) translation: (x, y) ↦ (\boxed{ }, \boxed{ })

Explanation:

Step1: Determine Reflection Rule

To find the reflection, we can compare a point from the original pentagon (e.g., \( W(0, 2) \)) and its image after reflection (before translation). Looking at the blue pentagon, let's track a point. Let's take \( W(0, 2) \). After reflection, before translation, let's see the coordinates. Wait, actually, the final image is after reflection and then translation. Let's take a point, say \( W(0, 2) \) and its image \( W'(4, -5) \). But first, reflection: let's check the y - axis or x - axis. Let's take point \( T(-7, 5) \) and its image after reflection (before translation). Wait, maybe better to check the reflection over the x - axis or y - axis. Wait, looking at the original (yellow) and the blue (after reflection and translation). Let's take point \( W(0, 2) \). After reflection, let's assume reflection over the x - axis: \( (x,y)\to(x, - y) \). Then \( W(0,2)\to(0, - 2) \). Then translation right 4 and down 3: \( (0 + 4,-2-3)=(4, - 5) \), which matches \( W'(4, - 5) \). Let's check another point: \( V(1,5) \). Reflection over x - axis: \( (1, - 5) \), then translation: \( (1 + 4,-5 - 3)=(5, - 8) \)? Wait, no, \( V' \) is at \( (5, - 8) \)? Wait, the blue \( V' \) is at (5, - 8)? Wait, looking at the graph, \( V' \) is at (5, - 8)? Wait, maybe reflection over the x - axis. Wait, let's check \( U(-3,6) \). Reflection over x - axis: \( (-3,-6) \), then translation: \( (-3 + 4,-6-3)=(1, - 9) \), which matches \( U'(1, - 9) \). Yes! So reflection is over the x - axis: \( (x,y)\to(x, - y) \).

Step2: Determine Translation Rule

The translation is given as right 4 units and down 3 units. The rule for translation right \( h \) units and down \( k \) units is \( (x,y)\to(x + h,y - k) \). Here, \( h = 4 \) and \( k = 3 \), so translation rule is \( (x,y)\to(x + 4,y - 3) \).

Answer:

Reflection: \((x, y) \mapsto (x, -y)\)
Translation: \((x, y) \mapsto (x + 4, y - 3)\)