Question

each transformation below starts with the point $(-2,3)$. drag the correct resulting coordinates into the matching transformation.
answer choices:
$(3,2)$ $(-2,-3)$ $(2,3)$
$(2,-3)$ $(-3,-2)$
$(2,1)$
drag the answer choices to the correct category over here!
translation 4 units right, 2 units down
reflection across the $x$-axis
reflection across the $y$-axis
rotation $90^{circ}$ clockwise
rotation $90^{circ}$ counterclockwise
rotation $180^{circ}$ about the origin

Explanation:

Step1: Translation 4 units right, 2 units down

For a translation \( (x,y)\to(x + 4,y-2) \). Given \( (x,y)=(-2,3) \), then \( x+4=-2 + 4=2 \), \( y - 2=3-2 = 1 \). The new coordinate is \( (2,1) \).

Step2: Reflection across the \(x\) - axis

For a reflection across the \(x\) - axis \( (x,y)\to(x,-y) \). Given \( (x,y)=(-2,3) \), then the new coordinate is \( (-2,-3) \).

Step3: Reflection across the \(y\) - axis

For a reflection across the \(y\) - axis \( (x,y)\to(-x,y) \). Given \( (x,y)=(-2,3) \), then \( -x=-(-2)=2 \), and the new coordinate is \( (2,3) \).

Step4: Rotation \(90^{\circ}\) clockwise

The rule for a \(90^{\circ}\) clockwise rotation \( (x,y)\to(y,-x) \). Given \( (x,y)=(-2,3) \), then \( y = 3\), \( -x=-(-2)=2 \). The new coordinate is \( (3,-2) \). Wait, no, the correct rule: if we have a point \( (x,y)\) and rotate \(90^{\circ}\) clockwise about the origin, the formula is \( (x,y)\to(y,-x)\). So \( (-2,3)\to(3,2) \).

Step5: Rotation \(90^{\circ}\) counter - clockwise

The rule for a \(90^{\circ}\) counter - clockwise rotation \( (x,y)\to(-y,x) \). Given \( (x,y)=(-2,3) \), then \( -y=-3\), \( x=-2 \). The new coordinate is \( (-3,-2) \).

Step6: Rotation \(180^{\circ}\) about the origin

The rule for a \(180^{\circ}\) rotation \( (x,y)\to(-x,-y) \). Given \( (x,y)=(-2,3) \), then \( -x=-(-2) = 2\), \( -y=-3 \). The new coordinate is \( (2,-3) \).

Answer:

• Translation 4 units right, 2 units down: \( (2,1) \)
• Reflection across the \(x\) - axis: \( (-2,-3) \)
• Reflection across the \(y\) - axis: \( (2,3) \)
• Rotation \(90^{\circ}\) clockwise: \( (3,2) \)
• Rotation \(90^{\circ}\) counter - clockwise: \( (-3,-2) \)
• Rotation \(180^{\circ}\) about the origin: \( (2,-3) \)