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 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 transforms a point $(x,y)$ to $(-y,x)$.

Step2: Analyze translation rules

A translation right 1 unit and down 2 units transforms a point $(x,y)$ to $(x + 1,y-2)$. A translation right 2 units and up 3 units transforms a point $(x,y)$ to $(x + 2,y + 3)$. A reflection across the $x$-axis transforms a point $(x,y)$ to $(x,-y)$.

Step3: Test the first option

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

Step4: Test the second option

Let the coordinates of a point on $\triangle IJK$ be $(x,y)$. After a translation right 2 units and up 3 units, the point becomes $(x + 2,y+3)$. Then after a reflection across the $x$-axis, the point becomes $(x + 2,-(y + 3))$.
Let's take a vertex of $\triangle IJK$, say $I(-8,5)$. After translation right 2 units and up 3 units, $I$ becomes $(-8 + 2,5 + 3)=(-6,8)$. After reflection across the $x$-axis, it becomes $(-6,-8)$ which is the coordinate of $I'$.
We can test other vertices $J(-7,2)$ and $K(-3,5)$ in the same way. For $J(-7,2)$, 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 coordinate of $J'$. For $K(-3,5)$, 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 coordinate of $K'$.

Answer:

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