Question

the graph shows triangles wxy and wxy. which sequence of transformations maps wxy onto wxy? a translation left 8 units and up 3 units followed by a reflection across the x - axis a reflection across the y - axis followed by a rotation 90° clockwise around the origin a rotation 180° around the origin followed by a translation left 4 units and down 3 units

Explanation:

Step1: Analyze translation rule

Translation left 8 units and up 3 units: $(x,y)\to(x - 8,y + 3)$. Then reflection across the $x$-axis: $(x,y)\to(x,-y)$. Let's assume a point $(x,y)$ on $\triangle WXY$. After translation, it becomes $(x - 8,y + 3)$, and after reflection across the $x$-axis, it becomes $(x - 8,-(y + 3))$.

Step2: Analyze reflection - rotation rule

Reflection across the $y$-axis: $(x,y)\to(-x,y)$. Rotation $90^{\circ}$ clockwise around the origin: $(x,y)\to(y,-x)$. If we start with $(x,y)$, after reflection across the $y$-axis we have $(-x,y)$, and after rotation $90^{\circ}$ clockwise we get $(y,x)$.

Step3: Analyze rotation - translation rule

Rotation $180^{\circ}$ around the origin: $(x,y)\to(-x,-y)$. Translation left 4 units and down 3 units: $(x,y)\to(x - 4,y-3)$. Starting with $(x,y)$, after rotation $180^{\circ}$ we have $(-x,-y)$, and after translation we get $(-x - 4,-y-3)$.
Let's assume a vertex of $\triangle WXY$ say $W(2,-8)$.

• For the first option: After translation left 8 units and up 3 units, $W(2,-8)\to(2 - 8,-8 + 3)=(-6,-5)$. After reflection across the $x$-axis, $(-6,-5)\to(-6,5)$.
• For the second option: After reflection across the $y$-axis, $W(2,-8)\to(-2,-8)$. After rotation $90^{\circ}$ clockwise around the origin, $(-2,-8)\to(-8,2)$.
• For the third option: After rotation $180^{\circ}$ around the origin, $W(2,-8)\to(-2,8)$. After translation left 4 units and down 3 units, $(-2,8)\to(-2-4,8 - 3)=(-6,5)$.

Let's check all vertices of the triangle. If we consider the general transformation rules for all vertices of $\triangle WXY$ and match with the vertices of $\triangle W'X'Y'$.
The sequence of a translation left 8 units and up 3 units followed by a reflection across the $x$-axis maps $\triangle WXY$ onto $\triangle W'X'Y'$.

Answer:

a translation left 8 units and up 3 units followed by a reflection across the $x$-axis