To determine the coordinate rule (composition of functions) that maps $\triangle DEF$ onto $\triangle D'E'F'$, let's analyze the transformation using the provided graph.

Step 1: Identify coordinates of $\triangle DEF$ and $\triangle D'E'F'$

From the graph:

• $D(-3,4)$, $E(-4,-3)$, $F(0,1)$
• $D'(3,4)$, $E'(4,-3)$, $F'(0,-1)$

Step 2: Analyze transformations

• Reflection over the $y$-axis: Reflecting $\triangle DEF$ across the $y$-axis, the coordinates of each point change sign in the $x$-coordinate: $D(-3,4) \to D'(3,4), \quad E(-4,-3) \to E'(4,-3), \quad F(0,1) \to F'(0,1)$

• Vertical reflection (across the $x$-axis): After reflection across the $y$-axis, reflect vertically across the $x$-axis: $D'(3,4) \to D'(3,4), \quad E'(4,-3) \to E'(4,-3), \quad F'(0,1) \to F'(0,-1)$

Step 3: Compose the transformation

The transformation rule involves two steps:

1. Reflect over the $y$-axis: $(x, y) \to (-x, y)$
2. Reflect over the $x$-axis: $(x, y) \to (x, -y)$

Combining these, the composite transformation rule is: $(x, y) \to (-x, -y)$

Final Answer:

$(x, y) \to (-x,-y)$

Would you like further details on reflection transformations or how to validate this step-by-step? Let me know!

Here are 5 related questions for further exploration:

1. How do you verify if two triangles are congruent using transformations?
2. What other transformations could map a triangle onto a congruent triangle?
3. How does a composite transformation differ from a single transformation?
4. What is the difference between rotation and reflection as transformations?
5. Can we describe transformations in terms of matrices?

Tip: Always verify transformations by applying the rule to all vertices and checking if they align with the image triangle.