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The problem shows a transformation of the basic quadratic function \( f(x) = x^2 \). The basic graph is a parabola opening upwards with its vertex at \( (0, 0) \).

In the given graph, the parabola is still opening upwards, but it appears to have shifted downwards along the \(y\)-axis. This type of transformation is called a **vertical shift**.

Looking at the graph:
- The vertex of the transformed graph is at \( (0, -3) \), which means the graph has been shifted 3 units **down**.

The rule for this transformation is:
\[
f(x) = x^2 - 3
\]

This represents a vertical shift of the function \( f(x) = x^2 \) by 3 units downward.

### Would you like more details or have any questions?

Here are 5 related questions to expand on this concept:
1. What happens to the graph of \( f(x) = x^2 \) if we shift it upwards by 2 units?
2. How would the graph change if we applied a horizontal shift to \( f(x) = x^2 \)?
3. What does multiplying \( f(x) = x^2 \) by a negative number do to the graph?
4. How does a vertical stretch affect the shape of the parabola \( f(x) = x^2 \)?
5. Can you describe the transformation if the function was \( f(x) = (x - 2)^2 + 1 \)?

**Tip**: Shifts along the \(x\)- and \(y\)-axes in a function like \( f(x) = x^2 \) can be quickly determined by analyzing changes in the vertex location and direction of the parabola.

Use transformations on the basic quadratic function f(x) = x^2 to write a rule y = f(x) that would produce the given graph. The graph shows a vertical shift of the parabola downwards.

Learn how to apply vertical shifts to the quadratic function f(x) = x^2. This problem explains how to shift the graph of a parabola and write the transformation rule for the given graph. Suitable for students in Grades 9-11.

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