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.