To determine the domain and range of the parabola defined by the function [tex]\( f(x) = x^2 + 6x + 5 \)[/tex], let's break down the problem step by step.
### Step 1: Determine the Domain
For any quadratic function of the form [tex]\( f(x) = ax^2 + bx + c \)[/tex], the domain is all real numbers. This is because you can input any real number for [tex]\( x \)[/tex] and get a corresponding [tex]\( f(x) \)[/tex].
Domain: all real numbers
### Step 2: Determine the Range
The range of a quadratic function depends on the direction the parabola opens and the vertex of the parabola.
1. Identify the direction: Since the leading coefficient [tex]\( a \)[/tex] (which is 1 in this case) is positive, the parabola opens upwards. Therefore, the minimum value of [tex]\( f(x) \)[/tex] occurs at the vertex.
2. Find the vertex: The vertex form of the quadratic equation [tex]\( f(x) = ax^2 + bx + c \)[/tex] helps in identifying the vertex. The x-coordinate of the vertex can be found using the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
Plugging in values [tex]\( a = 1 \)[/tex] and [tex]\( b = 6 \)[/tex]:
[tex]\[
h = -\frac{6}{2 \cdot 1} = -3
\][/tex]
3. Find the y-coordinate of the vertex: Substitute [tex]\( h \)[/tex] back into the function to find the minimum value [tex]\( k \)[/tex]:
[tex]\[
k = f(-3)
\][/tex]
Substitute [tex]\( x = -3 \)[/tex] into [tex]\( f(x) = x^2 + 6x + 5 \)[/tex]:
[tex]\[
f(-3) = (-3)^2 + 6(-3) + 5 = 9 - 18 + 5 = -4
\][/tex]
The vertex of the parabola is at [tex]\( (-3, -4) \)[/tex].
Since the parabola opens upwards, the range is all values of [tex]\( f(x) \)[/tex] that are greater than or equal to the minimum value [tex]\( -4 \)[/tex].
Range: [tex]\( f(x) \geq -4 \)[/tex]
### Conclusion
Now that we've determined both the domain and range, we can select the correct answer.
Domain is all real numbers. Range is [tex]\( f(x) \geq -4 \)[/tex].
