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**2. Identifica cuál de las gráficas corresponde a la función f(x) = -3x² + 6x, determina su eje de simetría y si su vértice es un máximo o un mínimo.**

Let's solve this step-by-step:

1. **Identify the type of function:**
   The given function is \( f(x) = -3x^2 + 6x \). This is a quadratic function in the form \( ax^2 + bx + c \), where \( a = -3 \), \( b = 6 \), and \( c = 0 \).

2. **Determine the axis of symmetry:**
   For a quadratic function \( ax^2 + bx + c \), the axis of symmetry is given by the formula:

   \[
   x = -\frac{b}{2a}
   \]

   Substituting the values of \( a = -3 \) and \( b = 6 \):

   \[
   x = -\frac{6}{2(-3)} = -\frac{6}{-6} = 1
   \]

   So, the axis of symmetry is \( x = 1 \).

3. **Find the vertex:**
   The vertex of the parabola can be found by plugging the x-coordinate of the axis of symmetry into the function:

   \[
   f(1) = -3(1)^2 + 6(1) = -3 + 6 = 3
   \]

   So, the vertex of the parabola is at \( (1, 3) \).

4. **Determine if the vertex is a maximum or a minimum:**
   Since the coefficient \( a = -3 \) is negative, the parabola opens downwards. This means the vertex represents a maximum point.

### Conclusion:
- The axis of symmetry is \( x = 1 \).
- The vertex is at \( (1, 3) \).
- The vertex is a maximum.

Would you like more details on how to graph this function or analyze other aspects of it? Here are some related questions that may help expand your understanding:

1. How do you find the y-intercept of a quadratic function?
2. How does the sign of the coefficient 'a' affect the shape of a quadratic graph?
3. What is the significance of the axis of symmetry in quadratic functions?
4. How can you determine the roots of the quadratic function \( f(x) = -3x^2 + 6x \)?
5. What is the process to convert a quadratic function into its vertex form?

**Tip:** To better visualize the function, try sketching the parabola using its vertex, axis of symmetry, and intercepts!

Identifica cuál de las gráficas corresponde a la función f(x) = -3x² + 6x, determina su eje de simetría y si su vértice es un máximo o un mínimo.

Learn how to solve the quadratic function f(x) = -3x² + 6x by determining the axis of symmetry and identifying whether the vertex is a maximum or minimum. This step-by-step solution involves key concepts in quadratic functions and algebra.

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