What is the image point of left parenthesis, minus, 9, comma, minus, 2, right parenthesis(−9,−2) after the transformation r, start subscript, x-axis, end subscript, circle, T, start subscript, minus, 1, comma, 3, end subscriptr 
x-axis
​
 ∘T 
−1,3
​
 ?

Let's break down the transformation step by step:

### 1. Translation \( T_{(-1, 3)} \):
The translation \( T_{(-1, 3)} \) means shifting the point by \(-1\) unit along the \(x\)-axis and \(+3\) units along the \(y\)-axis.

For the point \((-9, -2)\):

- The new \(x\)-coordinate will be \( -9 + (-1) = -10 \).
- The new \(y\)-coordinate will be \( -2 + 3 = 1 \).

So after the translation, the point becomes \((-10, 1)\).

### 2. Reflection across the \(x\)-axis \( r_{x-axis} \):
To reflect a point across the \(x\)-axis, the \(x\)-coordinate remains the same, while the \(y\)-coordinate changes its sign.

For the point \((-10, 1)\):

- The \(x\)-coordinate remains \( -10 \).
- The \(y\)-coordinate becomes \( -1 \).

Thus, after the reflection, the point is \((-10, -1)\).

### Final Answer:
The image of the point \((-9, -2)\) after the transformation \( r_{x-axis} \circ T_{(-1, 3)} \) is \((-10, -1)\).

Would you like further details or have any questions? Here are five related questions for further exploration:

1. How would the transformation change if the reflection was across the \(y\)-axis instead?
2. What is the matrix representation of a reflection across the \(x\)-axis?
3. How do you calculate the inverse of a translation transformation?
4. What happens if you apply the transformations in reverse order?
5. How does the slope of a line connecting the original and final points change after these transformations?

**Tip**: When reflecting across the \(x\)-axis, only the \(y\)-coordinate changes, while the \(x\)-coordinate remains unchanged.

What is the image point of (−9,−2) after the transformation r_x-axis ∘ T_{(-1,3)}?

Learn how to find the image of the point (−9,−2) after applying a translation of (-1,3) followed by a reflection across the x-axis. This step-by-step solution covers key concepts like translations and reflections, suitable for students in Grades 9-10.

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