https://r2.erweima.ai/mathbot/image/1724888900291gi2j3n02.jpg

La  función de la gráfica es: Seleccione una:

a.
y= 2x-2


b.
y=(x+2)2


c.
y= x+2


d.
y= 3x2-2

To determine the correct function from the options given, we need to identify the function that matches the shape and position of the parabola in the graph.

### Analysis of the Graph:
- The graph shows a parabola opening upwards.
- The vertex of the parabola is at the point \((-2, 0)\).
- The parabola passes through the points \((-2, 0)\), \((0, 4)\), and \((-4, 4)\).

### Matching the Functions:
Let's check each function:

1. **Option a: \(y = 2x - 2\)**
   - This is a linear function, not a parabola, so it cannot be correct.

2. **Option b: \(y = (x + 2)^2\)**
   - This represents a parabola with a vertex at \((-2, 0)\), and it opens upwards.
   - When \(x = 0\), \(y = (0 + 2)^2 = 4\), which matches the point \((0, 4)\).
   - When \(x = -4\), \(y = (-4 + 2)^2 = 4\), which matches the point \((-4, 4)\).

3. **Option c: \(y = x + 2\)**
   - This is a linear function, not a parabola, so it cannot be correct.

4. **Option d: \(y = 3x^2 - 2\)**
   - This represents a parabola, but it has its vertex at \((0, -2)\), which does not match the graph.

### Conclusion:
The function that correctly represents the given graph is **Option b: \(y = (x + 2)^2\)**.

Would you like more details or have any questions about this?

Here are five related questions to deepen your understanding:

1. How do you determine the vertex of a parabola from its equation?
2. What are the characteristics of the standard form of a quadratic function?
3. How does the graph of a parabola change when you modify the coefficient of \(x^2\)?
4. What is the significance of the vertex form of a parabola?
5. How can you find the axis of symmetry for a quadratic function?

**Tip:** When identifying a parabola's equation from a graph, always look for the vertex's coordinates and the direction in which the parabola opens (upward or downward).

Learn how to identify the correct quadratic function from a graph using detailed analysis. This problem involves analyzing parabolic graphs, vertex form of quadratic functions, and matching graphical characteristics with function equations. Suitable for high school students.

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