Question

the rule r_y - axis ∘ r_0, 90°(x, y) is applied to δabc. which triangle shows the final image? 1 2 3 4

Explanation:

Step1: First, perform 90 - degree rotation

The rule for a 90 - degree counter - clockwise rotation about the origin $R_{0,90^{\circ}}(x,y)=(-y,x)$.

Step2: Then, perform y - axis reflection

The rule for a reflection over the y - axis $r_{y - axis}(x,y)=(-x,y)$. Combining the two transformations, if we start with a point $(x,y)$, after $R_{0,90^{\circ}}$ we get $(-y,x)$, and after $r_{y - axis}$ on $(-y,x)$ we get $(y,x)$.
If we apply these transformations to the vertices of $\triangle ABC$ and compare with the given triangles on the grid, we can determine the final image.

Answer:

(Without seeing the specific coordinates of $\triangle ABC$ and detailed properties of triangles 1 - 4, we can't give a definite choice. But the general way to solve is as above. If we assume we have done the transformation on vertices of $\triangle ABC$ and compared), we would choose the correct triangle among 1, 2, 3, 4 based on the above - derived transformation rule $(y,x)$ for each vertex.)