The question asks to identify the equation that represents the curve shown in the graph. This graph appears to be a parabola.

Let's analyze the graph step by step:

Observing the key points:

• The vertex of the parabola appears to be at $(2, -3)$.
• The parabola opens upwards, meaning the coefficient of $x^2$ is positive.

Standard form of a parabola:

The general vertex form of a parabola is:

$y = a(x - h)^2 + k$

Where:

• $(h, k)$ is the vertex.
• $a$ determines the direction and width of the parabola.

Since the vertex is at $(2, -3)$, the equation will have $h = 2$ and $k = -3$, leading to the equation:

$y = (x - 2)^2 - 3$

This matches the first option: $y = (x - 2)^2 - 3$.

Final Answer:

The correct equation representing the curve is $y = (x - 2)^2 - 3$.

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

Here are 5 related questions to expand your understanding:

1. How do you find the vertex of a parabola from its equation?
2. What role does the value of $a$ in $y = a(x - h)^2 + k$ play in determining the parabola's shape?
3. How can you convert a quadratic equation from standard form to vertex form?
4. What is the axis of symmetry for a parabola, and how can you find it?
5. How would the equation change if the parabola opened downwards?

Tip: When analyzing a quadratic function, the vertex form is often the most useful for quickly identifying the vertex and the direction of the parabola.