Question

1. which transformations would map the triangle below into quadrant 4?○ rotate about the origin 90° counter-clockwise and then translate down 2 units.○ reflect across the x-axis and then reflect across the y-axis.○ rotate about the origin 180° clockwise and then reflect across the y-axis.○ translate up 5 units and then translate down 2 units.

Explanation:

Step1: Identify triangle vertices

Let the triangle's vertices be $(1,4)$, $(3,2)$, $(3,5)$ (from the graph).

Step2: Test Option 1 (Rotate 90° CCW + translate down 2)

90° counter-clockwise rotation rule: $(x,y)\to(-y,x)$

• $(1,4)\to(-4,1)$; translate down 2: $(-4,1-2)=(-4,-1)$ (Quadrant III)
• $(3,2)\to(-2,3)$; translate down 2: $(-2,3-2)=(-2,1)$ (Quadrant II)
• $(3,5)\to(-5,3)$; translate down 2: $(-5,3-2)=(-5,1)$ (Quadrant II)

Result not in Quadrant IV.

Step3: Test Option 2 (Reflect x-axis + reflect y-axis)

Reflect x-axis: $(x,y)\to(x,-y)$; reflect y-axis: $(x,y)\to(-x,y)$
Combined: $(x,y)\to(-x,-y)$

• $(1,4)\to(-1,-4)$ (Quadrant III)
• $(3,2)\to(-3,-2)$ (Quadrant III)
• $(3,5)\to(-3,-5)$ (Quadrant III)

Result not in Quadrant IV.

Step4: Test Option 3 (Rotate 180° CW + reflect y-axis)

180° rotation rule: $(x,y)\to(-x,-y)$; reflect y-axis: $(x,y)\to(-x,y)$
Combined: $(x,y)\to(x,-y)$

• $(1,4)\to(1,-4)$ (Quadrant IV)
• $(3,2)\to(3,-2)$ (Quadrant IV)
• $(3,5)\to(3,-5)$ (Quadrant IV)

Result is in Quadrant IV.

Step5: Test Option 4 (Translate up 5 + down 2)

Net translation: up 3 units, $(x,y)\to(x,y+3)$

• $(1,4)\to(1,7)$ (Quadrant I)
• $(3,2)\to(3,5)$ (Quadrant I)
• $(3,5)\to(3,8)$ (Quadrant I)

Result not in Quadrant IV.

Answer:

Rotate about the origin 180° clockwise and then reflect across the y-axis.