To determine which function corresponds to the given graph, we will analyze the features of the parabola in the image.

Key Observations:

• The vertex of the parabola is at approximately $(1, 0)$, suggesting that the vertex form might be $y = (x - 1)^2$.
• The parabola opens upwards, indicating a positive coefficient for the $x^2$ term.
• The y-intercept is at $(0, 1)$.

Now, let’s analyze the options:

• Option a) $x^2 + 2x - 4$: This would have a vertex at $x = -1$, not $x = 1$, so this is incorrect.
• Option b) $x^2 + 4x + 4$: The vertex of this function is at $x = -2$, which is incorrect.
• Option c) $2x^2 - 3x + 1$: This has the correct y-intercept at $(0, 1)$, but we need to check the vertex.
• Option d) $x^2 + 2x + 4$: This function has a vertex at $x = -1$, which doesn’t match the graph.

Calculation Check for Option c:

The quadratic formula shows the vertex of $2x^2 - 3x + 1$ is located at: $x_{\text{vertex}} = \frac{-(-3)}{2(2)} = \frac{3}{4} \approx 1$ This is consistent with the vertex at $(1, 0)$.

Thus, the correct function is: $\boxed{2x^2 - 3x + 1}$

Would you like further details or have any questions?

Here are 5 follow-up questions to explore:

1. How do you find the vertex of a quadratic function?
2. How can you determine if a parabola opens upwards or downwards?
3. What is the importance of the discriminant in quadratic equations?
4. How do transformations affect the graph of a parabola?
5. What are the steps to find the roots of a quadratic equation?

Tip: The vertex of a quadratic function in the form $ax^2 + bx + c$ is given by $x = -\frac{b}{2a}$.