Question

the graph shows triangles ijk and ijk. which sequence of transformations maps ijk onto ijk? a rotation 90° counterclockwise around the origin followed by a translation right 1 unit and down 2 units a reflection across the y - axis followed by a rotation 180° around the origin a translation right 2 units and up 3 units followed by a reflection across the x - axis

Explanation:

Step1: Analyze rotation rules

A 90 - degree counter - clockwise rotation around the origin has the rule $(x,y)\to(-y,x)$. Then a translation right 1 unit and down 2 units has the rule $(x,y)\to(x + 1,y-2)$. Let's assume a point $(x,y)$ on $\triangle IJK$. After 90 - degree counter - clockwise rotation, it becomes $(-y,x)$. After the translation, it becomes $(-y + 1,x-2)$. This does not map $\triangle IJK$ to $\triangle I'J'K'$.

Step2: Analyze reflection and rotation rules

A reflection across the $y$ - axis has the rule $(x,y)\to(-x,y)$. Then a 180 - degree rotation around the origin has the rule $(x,y)\to(-x,-y)$. If we start with a point $(x,y)$ on $\triangle IJK$, after reflection across the $y$ - axis, it is $(-x,y)$. After 180 - degree rotation, it is $(x,-y)$. This does not map $\triangle IJK$ to $\triangle I'J'K'$.

Step3: Analyze translation and reflection rules

Let the coordinates of a point on $\triangle IJK$ be $(x,y)$. A translation right 2 units and up 3 units has the rule $(x,y)\to(x + 2,y + 3)$. Then a reflection across the $x$ - axis has the rule $(x,y)\to(x,-y)$. So the combined rule is $(x,y)\to(x + 2,-(y + 3))$.
Let's take point $I(-8,5)$ of $\triangle IJK$. After translation right 2 units and up 3 units, it becomes $(-8+2,5 + 3)=(-6,8)$. After reflection across the $x$ - axis, it becomes $(-6,-8)$ which is the coordinates of $I'$. Similarly, for point $J(-7,2)$, after translation right 2 units and up 3 units, it becomes $(-7 + 2,2+3)=(-5,5)$. After reflection across the $x$ - axis, it becomes $(-5,-5)$ which is the coordinates of $J'$. For point $K(-3,5)$, after translation right 2 units and up 3 units, it becomes $(-3+2,5 + 3)=(-1,8)$. After reflection across the $x$ - axis, it becomes $(-1,-8)$ which is the coordinates of $K'$.

Answer:

a translation right 2 units and up 3 units followed by a reflection across the $x$-axis