https://file.aiquickdraw.com/mathbot/file/1730412431868az6e4wi7.jpg

To solve this problem, we start by understanding the transformation of the function due to a vector translation.

The original equation of the graph is:
\[
y = x^2
\]

The translation vector provided is:
\[
\begin{pmatrix} 0 \\ -2 \end{pmatrix}
\]

### Explanation of Translation
The vector \(\begin{pmatrix} 0 \\ -2 \end{pmatrix}\) indicates a vertical shift:
- Since the x-component is 0, there is no horizontal movement.
- The y-component is -2, meaning the graph moves 2 units down.

### Translating the Equation
To translate the function \(y = x^2\) down by 2 units, we subtract 2 from the entire function:
\[
y = x^2 - 2
\]

### Solution
The translated equation is:
\[
y = x^2 - 2
\]

Looking at the options, this matches with the circled answer on the paper:
\[
y = x^2 - 2
\]

This is the correct equation for the translated graph.

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:
1. What would the equation be if the translation vector was \(\begin{pmatrix} 0 \\ +3 \end{pmatrix}\)?
2. How would the equation change if the translation vector had a horizontal component (e.g., \(\begin{pmatrix} 3 \\ -2 \end{pmatrix}\))?
3. How does the graph of \(y = x^2 - 2\) differ visually from \(y = x^2\)?
4. What is the general form for translating any function by a vector \(\begin{pmatrix} a \\ b \end{pmatrix}\)?
5. How does translating a quadratic function affect its vertex?

**Tip:** When translating graphs, horizontal translations affect the input \(x\), while vertical translations affect the output \(y\).

The graph with equation y = x^2 is translated by vector (0, -2). Circle the equation of the translated graph.

Learn how to translate the quadratic graph y = x^2 by a vector (0, -2). This example shows a vertical shift in the function, resulting in the translated equation y = x^2 - 2. Suitable for students in Grades 8-10, covering graph translation concepts.

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