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The image shows two functions, \( h(x) = -x^2 \) and \( k(x) = 2x^2 + 8x + 8 \), along with multiple graph options and a set of function choices. Here's how to interpret and match the functions to the graphs:

1. **For \( h(x) = -x^2 \):**
   - This is a simple parabola opening downward because of the negative coefficient on \( x^2 \). The vertex is at the origin (0, 0).
   
2. **For \( k(x) = 2x^2 + 8x + 8 \):**
   - This is a quadratic equation, which can be factored or rewritten in vertex form. Let's complete the square:
     \[
     k(x) = 2(x^2 + 4x) + 8
     \]
     \[
     k(x) = 2((x+2)^2) \implies \text{vertex form: } k(x) = 2(x+2)^2
     \]
     - The vertex is at \( (-2, 0) \), and it opens upward because of the positive coefficient on \( x^2 \).

Next, match these functions to the graphs:

- \( h(x) = -x^2 \): The graph that opens downward with the vertex at (0, 0) matches the first function \( h(x) \).
- \( k(x) = 2(x + 2)^2 \): The graph that opens upward with a vertex at (-2, 0) matches the second function \( k(x) \).

Would you like a more detailed explanation of the graph transformation process or any further clarifications?

Here are five questions to deepen understanding:
1. How can you determine the direction in which a parabola opens based on the quadratic function?
2. What effect does the coefficient of \( x^2 \) have on the width of a parabola?
3. How do you find the vertex of a quadratic function in standard form?
4. What transformations occur when you complete the square for a quadratic function?
5. How can you tell from a graph whether a function has been shifted horizontally or vertically?

**Tip**: Completing the square is a useful technique to convert a quadratic function into vertex form, making it easier to graph.

Match the functions h(x) = -x^2 and k(x) = 2x^2 + 8x + 8 with their corresponding graphs.

Learn how to match quadratic functions like h(x) = -x^2 and k(x) = 2x^2 + 8x + 8 to their corresponding parabolas on the graph. This problem demonstrates graph transformations and vertex form of quadratic functions, suitable for students in Grades 8-10.

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