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The problem provided asks to determine the graph of \( h(x) = -(x - 5)^2 \) using transformations of the basic quadratic function \( f(x) = x^2 \).

### Steps to determine the graph of \( h(x) \):

1. **Start with the basic graph of \( f(x) = x^2 \):**
   - This is a parabola with its vertex at the origin \((0, 0)\), opening upwards.

2. **Horizontal shift:**
   - The expression \( (x - 5)^2 \) indicates a **horizontal shift**.
   - The graph of \( x^2 \) is shifted **5 units to the right**. So, the new vertex is at \((5, 0)\).

3. **Reflection over the x-axis:**
   - The negative sign in front of the function, \( -(x - 5)^2 \), reflects the graph across the x-axis.
   - This means that instead of opening upwards, the parabola now opens **downwards**.

### Final graph:
- The graph of \( h(x) = -(x - 5)^2 \) is a parabola with its vertex at \((5, 0)\), opening downward.

Would you like more details on any step?

Here are five related questions for practice:
1. What is the vertex form of a quadratic equation?
2. How do you determine the direction a parabola opens?
3. How does a vertical stretch or compression affect a parabola?
4. What happens to the graph if the horizontal shift is left instead of right?
5. How would adding a constant \( c \) to \( -(x - 5)^2 + c \) affect the graph?

**Tip:** Always look for shifts, reflections, and stretches when transforming graphs!

Use transformations of the graph of f(x) = x^2 to determine the graph of the given function: h(x) = -(x - 5)^2.

Learn how to transform the graph of the quadratic function f(x) = x^2 to graph h(x) = -(x - 5)^2. This tutorial covers key transformations, including horizontal shifts and reflections. Ideal for students in Grades 9-11.

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