Students match graphs of parabolas to their corresponding function transformations, such as vertical shifts and reflections.

To solve the problem, we need to match each graph of a quadratic function with its corresponding equation. Quadratic functions are typically written in the form \( f(x) = a(x-h)^2 + k \), where:

- \( (h, k) \) is the vertex of the parabola.
- \( a \) determines the direction and the width of the parabola:
- If \( a > 0 \), the parabola opens upwards.
- If \( a < 0 \), the parabola opens downwards.
- The absolute value of \( a \) affects the width: larger \( |a| \) makes the parabola narrower, and smaller \( |a| \) makes it wider.

Let's analyze each graph and match it with the correct equation.

Graph A

- The parabola opens upwards.
- The vertex is at \( (3, -2) \).
- The equation that matches this is \( f(x) = (x-3)^2 - 2 \).

Match: A → I

Graph B

- The parabola opens upwards.
- The vertex is at \( (5, 0) \).
- The equation that matches this is \( f(x) = (x-5)^2 \).

Match: B → F

Graph C

- The parabola opens upwards.
- The vertex is at \( (4, 0) \).
- The equation that matches this is \( f(x) = (x-4)^2 \).

Match: C → D

Graph D

- The parabola opens upwards.
- The vertex is at \( (2, 3) \).
- The equation that matches this is \( f(x) = (x-2)^2 + 3 \).

Match: D → E

Graph E

- The parabola opens upwards.
- The vertex is at \( (5, 0) \).
- The equation that matches this is \( f(x) = (x-5)^2 \).

Match: E → F

Graph F

- The parabola opens upwards.
- The vertex is at \( (4, 0) \).
- The equation that matches this is \( f(x) = (x-4)^2 \).

Match: F → D

Graph G

- The parabola opens upwards.
- The vertex is at \( (0, 0) \).
- The equation that matches this is \( f(x) = x^2 \).

Match: G → H

Graph H

- The parabola opens upwards.
- The vertex is at \( (0, 0) \).
- The equation that matches this is \( f(x) = x^2 \).

Match: H → G

Graph I

- The parabola opens upwards.
- The vertex is at \( (3, -2) \).
- The equation that matches this is \( f(x) = (x-3)^2 - 2 \).

Match: I → A

Graph J

- The parabola opens downwards.
- The vertex is at \( (0, 0) \).
- The equation that matches this is \( f(x) = -x^2 \).

Match: J → L

Graph K

- The parabola opens upwards.
- The vertex is at \( (0, 0) \).
- The equation that matches this is \( f(x) = x^2 \).

Match: K → H

Graph L

- The parabola opens downwards.
- The vertex is at \( (0, 0) \).
- The equation that matches this is \( f(x) = -x^2 \).

Match: L → J

Final Answer

\[
\boxed{
\begin{array}{ll}
A \to I & B \to F \\
C \to D & D \to E \\
E \to F & F \to D \\
G \to H & H \to G \\
I \to A & J \to L \\
K \to H & L \to J \\
\end{array}
}
\]