To find the image point of $(0,7)$ after the transformation $T_{3,0} \circ r_{y=-x}$, let's break down the process step by step.

Step 1: Apply the reflection $r_{y=-x}$

The transformation $r_{y=-x}$ reflects a point across the line $y = -x$. The rule for this reflection is: $(x, y) \rightarrow (-y, -x)$ So, applying this reflection to the point $(0, 7)$, we get: $(0, 7) \rightarrow (-7, 0)$

Step 2: Apply the translation $T_{3,0}$

The translation $T_{3,0}$ moves a point 3 units to the right and 0 units up. The rule for this transformation is: $(x, y) \rightarrow (x + 3, y + 0)$ So, applying this translation to the point $(-7, 0)$, we get: $(-7, 0) \rightarrow (-7 + 3, 0 + 0) = (-4, 0)$

Final Answer:

The image point of $(0, 7)$ after the transformation $T_{3,0} \circ r_{y=-x}$ is $(-4, 0)$.

Would you like any more details or clarifications on this?

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Here are 5 related questions:

1. What is the reflection across the line $y = x$?
2. How would the point change if we reflect it across the x-axis?
3. Can a transformation be a combination of reflection and rotation?
4. How would the point $(3,4)$ transform under $T_{5,2} \circ r_{y=-x}$?
5. What is the result of translating a point by $(-2, -3)$?

Tip: When working with multiple transformations, apply them in the correct order—this can affect the final result!