Looking closely at the uploaded image, I’ll go over the key points to verify and refine the quadratic equation based on the graph.

Step-by-Step Analysis:

1. Vertex Identification:

   • The vertex of the parabola appears to be located at $(0, 2)$. This observation implies the vertex form of the equation would be centered around this point, making $h = 0$ and $k = 2$.

2. Using the Vertex Form:

   • The vertex form of a parabola is: $y = a(x - h)^2 + k$ Since $h = 0$ and $k = 2$, we start with: $y = ax^2 + 2$

3. Finding the Value of $a$:

   • To determine the value of $a$, let's find another point on the graph. Observing the graph, when $x = 1$, $y = 3$.
   • Substitute $x = 1$ and $y = 3$ into the equation to solve for $a$: $3 = a(1)^2 + 2$ $3 = a + 2$ $a = 1$

4. Formulate the Final Equation:

   • With $a = 1$, the equation of the quadratic function is: $y = x^2 + 2$

Final Answer:

The equation for the graphed quadratic function is: $y = x^2 + 2$

Would you like more details, or do you have any additional questions?

Here are five follow-up questions to explore this further:

1. How would the equation change if the parabola had a different vertex?
2. If the parabola opened downward, what would that mean for the value of $a$?
3. How can you verify this equation by checking other points on the graph?
4. What does changing the value of $k$ (in $y = x^2 + k$) do to the graph's position?
5. Could you use the standard form $y = ax^2 + bx + c$ directly from points if the vertex wasn’t obvious?

Tip: Double-check the vertex and direction of opening when determining the equation of a graphed parabola. These features give critical clues about the function's structure.