https://file.aiquickdraw.com/mathbot/file/1726173562408gv3vfwkx.jpg

The problem requires using transformations of the basic quadratic function \( f(x) = x^2 \) to match the new graph shown.

### Observations:
- The basic quadratic function \( f(x) = x^2 \) has a vertex at \( (0, 0) \) and opens upwards.
- In the given transformed graph, the vertex has shifted downward, and the new graph opens upwards as well, but it is vertically stretched.

### Step-by-step analysis:
1. **Vertical Shift**: The vertex in the new graph is at \( (0, -6) \) rather than \( (0, 0) \). This indicates a downward shift by 6 units.
   - So, the transformation includes: \( f(x) = x^2 - 6 \).

2. **Vertical Stretch**: The graph appears narrower, which suggests a vertical stretch. The basic quadratic function passes through points like \( (-1, 1) \), but in the transformed graph, this point seems to be stretched to higher values.
   - We need to multiply the function by a constant \( a \) to account for the stretch.

Let’s estimate the stretch factor using one of the points from the graph.

- From the table, at \( x = 1 \), the value is \( f(x) = 1 \).
- In the transformed graph, at \( x = 1 \), the value seems to be about \( 3 \).

Thus, the stretch factor \( a \) satisfies:
\[
a(1^2) - 6 = 3 \quad \Rightarrow \quad a - 6 = 3 \quad \Rightarrow \quad a = 9
\]

### Conclusion:
The transformation involves a vertical stretch by a factor of 9 and a downward shift by 6 units. Thus, the rule that would produce the given graph is:
\[
f(x) = 9x^2 - 6
\]

Would you like further details or have any questions about the transformations?

Here are 5 relative questions to deepen your understanding:
1. How would the graph look if the quadratic function were horizontally shifted?
2. What would happen if we applied a reflection over the x-axis to this graph?
3. Can you explain how a vertical compression would affect the graph?
4. How does the value of \( a \) in the equation \( f(x) = ax^2 \) impact the shape of the parabola?
5. What if the transformation included both horizontal and vertical shifts?

**Tip**: When applying transformations, always handle shifts first (up/down or left/right), then apply stretches or compressions, and lastly reflections if necessary.

Use transformations on the basic quadratic function f(x) = x^2 to write a rule y = f(x) that would produce the given graph.

Learn how to transform the basic quadratic function f(x) = x^2 to match a new graph. This problem involves vertical shifts and stretches, making it suitable for students in Grades 9-12.

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