https://r2.erweima.ai/mathbot/image/1726157307557x8kzfvf9.jpg

The image contains a set of quadratic functions where you are asked to find their roots (or zeros), domain, range, and finally plot each function. Let's address each function one by one:

### 1. \( P(x) = x^2 - 6x + 9 \)
- **Roots (Zeros)**: This is a perfect square trinomial: \( P(x) = (x - 3)^2 \). So, the root is \( x = 3 \).
- **Domain**: The domain of any quadratic function is \( (-\infty, \infty) \).
- **Range**: Since the parabola opens upwards (positive \(x^2\) term) and the vertex is at \( x = 3 \), the minimum value is \( y = 0 \). So, the range is \( [0, \infty) \).
  
### 2. \( P(x) = 3x^2 + 5x - 2 \)
- **Roots (Zeros)**: We use the quadratic formula:
  \[
  x = \frac{-5 \pm \sqrt{5^2 - 4(3)(-2)}}{2(3)} = \frac{-5 \pm \sqrt{25 + 24}}{6} = \frac{-5 \pm \sqrt{49}}{6}
  \]
  So the roots are \( x = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3} \) and \( x = \frac{-5 - 7}{6} = \frac{-12}{6} = -2 \).
- **Domain**: \( (-\infty, \infty) \)
- **Range**: The parabola opens upwards, and the range is \( [y_{\text{min}}, \infty) \), where \( y_{\text{min}} \) is the vertex’s y-coordinate.

### 3. \( P(x) = -x^2 + 5x - 4 \)
- **Roots (Zeros)**: Again, using the quadratic formula:
  \[
  x = \frac{-5 \pm \sqrt{5^2 - 4(-1)(-4)}}{2(-1)} = \frac{-5 \pm \sqrt{25 - 16}}{-2} = \frac{-5 \pm \sqrt{9}}{-2}
  \]
  The roots are \( x = \frac{-5 + 3}{-2} = 1 \) and \( x = \frac{-5 - 3}{-2} = 4 \).
- **Domain**: \( (-\infty, \infty) \)
- **Range**: Since the parabola opens downwards (negative \(x^2\) term), the maximum value occurs at the vertex.

### 4. \( P(x) = 4x^2 + 2 \)
- **Roots (Zeros)**: This function has no real roots because the discriminant is negative (no real solutions).
- **Domain**: \( (-\infty, \infty) \)
- **Range**: Since the parabola opens upwards, and the minimum value is at the vertex, the range is \( [2, \infty) \).

### 5. \( P(x) = 2x^2 + 5x - 3 \)
- **Roots (Zeros)**: Using the quadratic formula:
  \[
  x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)} = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm \sqrt{49}}{4}
  \]
  The roots are \( x = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2} \) and \( x = \frac{-5 - 7}{4} = \frac{-12}{4} = -3 \).
- **Domain**: \( (-\infty, \infty) \)
- **Range**: The parabola opens upwards, and the range is from the minimum value of the vertex.

### 6. \( P(x) = x^2 + 4x + 6 \)
- **Roots (Zeros)**: Using the quadratic formula:
  \[
  x = \frac{-4 \pm \sqrt{4^2 - 4(1)(6)}}{2(1)} = \frac{-4 \pm \sqrt{16 - 24}}{2}
  \]
  Since the discriminant is negative, there are no real roots.
- **Domain**: \( (-\infty, \infty) \)
- **Range**: The parabola opens upwards, and the minimum value is \( y_{\text{min}} = \text{vertex value} \).

### 7. \( P(x) = x^2 - 5x \)
- **Roots (Zeros)**: Factorize \( P(x) = x(x - 5) \). The roots are \( x = 0 \) and \( x = 5 \).
- **Domain**: \( (-\infty, \infty) \)
- **Range**: Since the parabola opens upwards, the range is from the vertex’s y-coordinate to infinity.

### 8. \( P(x) = -x^2 - 2x - 3 \)
- **Roots (Zeros)**: Using the quadratic formula:
  \[
  x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-1)(-3)}}{2(-1)} = \frac{2 \pm \sqrt{4 - 12}}{-2}
  \]
  There are no real roots since the discriminant is negative.
- **Domain**: \( (-\infty, \infty) \)
- **Range**: The parabola opens downwards, and the maximum value is at the vertex.

### 9. \( P(x) = x^2 - 5x - 4 \)
- **Roots (Zeros)**: Using the quadratic formula:
  \[
  x = \frac{5 \pm \sqrt{(-5)^2 - 4(1)(-4)}}{2(1)} = \frac{5 \pm \sqrt{25 + 16}}{2} = \frac{5 \pm \sqrt{41}}{2}
  \]
  The roots are \( x = \frac{5 \pm \sqrt{41}}{2} \), which are irrational.
- **Domain**: \( (-\infty, \infty) \)
- **Range**: The parabola opens upwards, and the range is from the vertex’s y-value.

Would you like a specific graph or further details on any of these functions?

Here are 5 questions to explore further:
1. How do you graph quadratic functions step by step?
2. What is the significance of the discriminant in the quadratic formula?
3. How do you find the vertex of a quadratic function algebraically?
4. How do the coefficients of a quadratic equation affect its graph?
5. Can quadratic functions have complex roots, and what does that mean geometrically?

**Tip**: The discriminant \( b^2 - 4ac \) tells you whether a quadratic has real roots (if positive), no real roots (if negative), or a single repeated root (if zero).

Encuentra sus raíces o ceros, dominio, rango y por último realiza su gráfica la función para las siguientes ecuaciones cuadráticas.

Explore how to find the roots, domain, range, and graph quadratic functions like x² - 6x + 9 and 3x² + 5x - 2. This problem covers essential algebra concepts, including using the quadratic formula and determining the vertex and range of parabolas.

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