Let's solve the problem step-by-step for the function [tex]\( f(x) = x^2 + 6x + 8 \)[/tex].
### 1. Axis of Symmetry
The axis of symmetry of a quadratic function given by [tex]\( f(x) = ax^2 + bx + c \)[/tex] is calculated using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Given:
[tex]\[
a = 1, \quad b = 6, \quad c = 8
\][/tex]
Substitute the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:
[tex]\[ x = -\frac{6}{2 \cdot 1} = -\frac{6}{2} = -3 \][/tex]
So, the axis of symmetry is:
[tex]\[ x = -3 \][/tex]
### 2. Vertex
The vertex of the quadratic function occurs at the axis of symmetry. To find the y-coordinate of the vertex, substitute [tex]\( x = -3 \)[/tex] back into the original function [tex]\( f(x) = x^2 + 6x + 8 \)[/tex]:
[tex]\[
f(-3) = (-3)^2 + 6(-3) + 8 = 9 - 18 + 8 = -1
\][/tex]
Therefore, the vertex is:
[tex]\[ (-3, -1) \][/tex]
### 3. y-Intercept
The y-intercept of the quadratic function is the value of [tex]\( f(x) \)[/tex] when [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the function [tex]\( f(x) = x^2 + 6x + 8 \)[/tex]:
[tex]\[
f(0) = 0^2 + 6 \cdot 0 + 8 = 8
\][/tex]
So, the y-intercept is:
[tex]\[ (0, 8) \][/tex]
### 4. Domain and Range
- Domain: The domain of any quadratic function is all real numbers because a quadratic function is defined for every real number [tex]\( x \)[/tex]. Hence, the domain is:
[tex]\[ (-\infty, \infty) \][/tex]
- Range: Since the parabola opens upwards (as [tex]\( a > 0 \)[/tex]), the range starts at the y-coordinate of the vertex and extends to infinity. The y-coordinate of the vertex is -1, so the range is:
[tex]\[ [-1, \infty) \][/tex]
### 5. Maximum or Minimum
The vertex represents either a maximum or a minimum value. Since the parabola opens upwards (because [tex]\( a > 0 \)[/tex]), the vertex is a minimum point. Therefore, the vertex is a:
[tex]\[ \text{minimum} \][/tex]
### Summary
- Axis of Symmetry: [tex]\( x = -3 \)[/tex]
- Vertex: [tex]\( (-3, -1) \)[/tex]
- y-Intercept: [tex]\( 8 \)[/tex]
- Domain: [tex]\( (-\infty, \infty) \)[/tex]
- Range: [tex]\( [-1, \infty) \)[/tex]
- Vertex is a minimum.
This completes the detailed step-by-step solution for the problem!
