On a coordinate plane, a parabola opens up. It goes through (negative 7, 5), has a vertex at (negative 4, negative 4), and goes through (negative 1, 5).

Question

The graph of the function f(x) = x2 + 8x + 12 is shown. Which statements describe the graph? Check all that apply.

The vertex is the maximum value.
The axis of symmetry is x = –4.
The domain is all real numbers.
The range is all real numbers.
The function is increasing over (–∞, –4).
The x-intercepts are at (–6, 0) and (–2, 0).

Answer 1

Let's analyze the given statements based on the properties of the parabola.

1. The vertex is the maximum value.
   This statement is false. Since the parabola opens upwards, the vertex represents the minimum value.

2. The axis of symmetry is x = –4.
   This statement is true. The axis of symmetry for a parabola in the form \(y = ax^2 + bx + c\) can be given by the formula \(x = -\frac{b}{2a}\). Since the vertex is at (-4, -4), the axis of symmetry is indeed \(x = -4\).

3. The domain is all real numbers.
   This statement is true. The domain of a quadratic function is always all real numbers.

4. The range is all real numbers.
   This statement is false. Since the parabola opens upwards and has a minimum value at the vertex, the range is \(y \geq -4\).

5. The function is increasing over (–∞, –4).
   This statement is true. In an upward-opening parabola, the function decreases to the left of the vertex and increases to the right. Therefore, it is indeed increasing for \(x > -4\).

6. The x-intercepts are at (–6, 0) and (–2, 0).
   This statement is true. To find the x-intercepts, you can set \(f(x) = 0\) and solve the quadratic equation. For the given quadratic \(f(x) = x^2 + 8x + 12\): \[ 0 = x^2 + 8x + 12 \] Factoring gives: \[ 0 = (x + 6)(x + 2) \] Hence, the x-intercepts are \(x = -6\) and \(x = -2\).

So the true statements are:

• The axis of symmetry is x = –4.
• The domain is all real numbers.
• The function is increasing over (–∞, –4).
• The x-intercepts are at (–6, 0) and (–2, 0).