Let's analyze the function [tex]\( f(x) = x^2 + 8x + 12 \)[/tex] and determine various properties of its graph step by step.
### 1. Vertex
The vertex of a quadratic function [tex]\( f(x) = ax^2 + bx + c \)[/tex] is given by the coordinates [tex]\((h, k)\)[/tex], where:
[tex]\[ h = -\frac{b}{2a} \][/tex]
[tex]\[ k = f(h) \][/tex]
Given [tex]\( a = 1 \)[/tex], [tex]\( b = 8 \)[/tex], and [tex]\( c = 12 \)[/tex]:
[tex]\[ h = -\frac{8}{2 \cdot 1} = -4 \][/tex]
[tex]\[ k = f(-4) = (-4)^2 + 8(-4) + 12 = 16 - 32 + 12 = -4 \][/tex]
So, the vertex is [tex]\((-4, -4)\)[/tex]. Since the parabola opens upwards (as [tex]\( a > 0 \)[/tex]), the vertex represents the minimum value of the function.
### 2. Axis of Symmetry
The axis of symmetry is a vertical line passing through the vertex, which is [tex]\( x = -4 \)[/tex].
### 3. Domain
The domain of any quadratic function is all real numbers. There are no restrictions on the values that [tex]\( x \)[/tex] can take.
### 4. Range
The range of the function is all real numbers greater than or equal to the [tex]\( k \)[/tex]-value of the vertex, which is [tex]\(-4\)[/tex]. Thus, the range is all real numbers greater than or equal to [tex]\(-4\)[/tex].
### 5. Increasing Interval
For the function [tex]\( f(x) = x^2 + 8x + 12 \)[/tex], the parabola opens upwards. The function decreases until it reaches the vertex [tex]\( x = -4 \)[/tex] and then increases. Therefore, the function is increasing over the interval [tex]\((-4, \infty)\)[/tex].
### 6. X-Intercepts
The x-intercepts are found by solving [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ x^2 + 8x + 12 = 0 \][/tex]
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{-8 \pm \sqrt{64 - 48}}{2} = \frac{-8 \pm \sqrt{16}}{2} = \frac{-8 \pm 4}{2} \][/tex]
This gives us:
[tex]\[ x = -2 \quad \text{or} \quad x = -6 \][/tex]
The x-intercepts are [tex]\((-2, 0)\)[/tex] and [tex]\((-6, 0)\)[/tex].
### Summary of Statements
- The vertex is the maximum value: False. The vertex is the minimum value because the parabola opens upwards.
- The axis of symmetry is [tex]\( x = -4 \)[/tex]: True.
- The domain is all real numbers: True.
- The range is all real numbers: False. The correct range is all real numbers greater than or equal to [tex]\(-4\)[/tex].
- The function is increasing over [tex]\((-6, -4)\)[/tex]: False. The function is actually increasing over [tex]\((-4, \infty)\)[/tex].
- The x-intercepts are at [tex]\((-6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex]: True.
### Correct Statements
- The axis of symmetry is [tex]\( x = -4 \)[/tex].
- The domain is all real numbers.
- The x-intercepts are at [tex]\((-6, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex].
