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You are asked to graph the following quadratic functions and determine their ranges. Let's first analyze the given functions:

1. \( f(x) = -x^2 - 4x + 12 \)
2. \( f(x) = x^2 - 6x + 7 \)

### 1.1. Function: \( f(x) = -x^2 - 4x + 12 \)

This is a quadratic equation in the form \( f(x) = ax^2 + bx + c \), where \( a = -1 \), \( b = -4 \), and \( c = 12 \).

- **Vertex**: The x-coordinate of the vertex is given by the formula \( x_v = \frac{-b}{2a} \). Substituting values:

  \[
  x_v = \frac{-(-4)}{2(-1)} = \frac{4}{-2} = -2
  \]

  To find the y-coordinate, substitute \( x_v = -2 \) back into the function:

  \[
  f(-2) = -(-2)^2 - 4(-2) + 12 = -4 + 8 + 12 = 16
  \]

  So, the vertex is \( (-2, 16) \).

- **Direction**: Since \( a = -1 \), the parabola opens downwards.

- **Range**: The maximum value occurs at the vertex, so the range is \( (-\infty, 16] \).

### 1.2. Function: \( f(x) = x^2 - 6x + 7 \)

This quadratic equation is in the form \( f(x) = ax^2 + bx + c \), where \( a = 1 \), \( b = -6 \), and \( c = 7 \).

- **Vertex**: The x-coordinate of the vertex is:

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

  Substituting \( x_v = 3 \) into the function:

  \[
  f(3) = (3)^2 - 6(3) + 7 = 9 - 18 + 7 = -2
  \]

  So, the vertex is \( (3, -2) \).

- **Direction**: Since \( a = 1 \), the parabola opens upwards.

- **Range**: The minimum value occurs at the vertex, so the range is \( [-2, \infty) \).

### Graphs and Range Summary:

- For \( f(x) = -x^2 - 4x + 12 \), the vertex is at \( (-2, 16) \), the parabola opens downwards, and the range is \( (-\infty, 16] \).
- For \( f(x) = x^2 - 6x + 7 \), the vertex is at \( (3, -2) \), the parabola opens upwards, and the range is \( [-2, \infty) \).

Let me know if you'd like to see the graphs or any further explanation.

### Follow-up Questions:

1. How can the discriminant help determine the number of real roots of each quadratic?
2. What is the axis of symmetry for each parabola?
3. How do the values of \( a \), \( b \), and \( c \) affect the shape and position of a quadratic function?
4. What happens to the graph if you increase or decrease the value of \( c \)?
5. How can you verify the vertex coordinates without using calculus?

### Tip:
To quickly identify whether a parabola opens upwards or downwards, check the sign of the \( a \)-coefficient: if it's positive, the parabola opens upwards; if negative, it opens downwards.

Graficar las siguientes ecuaciones cuadráticas y determinar el rango. 1.1 f(x) = -x^2 - 4x + 12, 1.2 f(x) = x^2 - 6x + 7

Learn how to graph the quadratic equations f(x) = -x^2 - 4x + 12 and f(x) = x^2 - 6x + 7. This problem covers algebraic concepts such as finding the vertex, determining the range, and analyzing the direction of parabolas. Suitable for students in Grades 9-11.

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