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.