https://file.aiquickdraw.com/mathbot/file/1736864589805qg7qu8nb.jpg

Thank you for sharing the image. Based on the instructions and your work in part 1, let's focus on **part 2** of the problem:  

**Part 2: Determine the equation of a quadratic relation that opens down and crosses the x-axis. Show all steps.**

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### Step-by-Step Solution

1. **General form of a quadratic relation (vertex form)**:
   \[
   y = a(x - h)^2 + k
   \]
   - \( (h, k) \) is the **vertex** of the parabola.
   - Since we need a downward-opening quadratic, **\( a < 0 \)**.

2. **From Part 1**, you determined:
   - Vertex = \( (0, 4) \).
   - The given equation was already:  
     \[
     y = -x^2 + 4.
     \]

3. **For part 2**, determine the x-intercepts:  
   To find where the parabola crosses the x-axis, set \( y = 0 \) and solve for \( x \).

   \[
   0 = -x^2 + 4.
   \]
   
   Rearranging:
   \[
   x^2 = 4.
   \]
   Taking the square root:
   \[
   x = \pm 2.
   \]

4. **Key points and equation confirmation**:
   - The vertex is at \( (0, 4) \).
   - The x-intercepts are at \( (-2, 0) \) and \( (2, 0) \), showing that the graph crosses the x-axis.
   - The negative \( a \) value causes the parabola to open downward, confirming that the equation is correct.

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### Final Equation
\[
y = -x^2 + 4.
\]

Would you like more clarification or steps on any part of this?  

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**Questions for further practice**:  
1. How do you convert a quadratic equation from vertex form to standard form?  
2. How do you find the axis of symmetry for a quadratic relation?  
3. What is the effect of changing the value of \( a \) in the quadratic equation?  
4. How can you find the maximum or minimum value of a quadratic function?  
5. What are the conditions for a quadratic function to have no x-intercepts?  

**Tip**: Always verify the direction of a parabola by checking the sign of the leading coefficient \( a \). Negative \( a \) opens downward, positive \( a \) opens upward.

Learn how to solve quadratic equations in vertex form, find the x-intercepts, and confirm the equation of a downward-opening parabola. This guide covers key steps and concepts, such as the vertex form y = a(x-h)^2 + k and solving for roots. Suitable for students in Grades 9-10.

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