Certainly! Let's analyze the quadratic function [tex]\( g(x) = x^2 + 4x + 3 \)[/tex] step by step.
### Step 1: Determine if the function has a minimum or maximum value
A quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex] will have a minimum value if [tex]\( a \)[/tex] is positive, and a maximum value if [tex]\( a \)[/tex] is negative.
Here, [tex]\( a = 1 \)[/tex]. Since [tex]\( a \)[/tex] is positive, the function has a minimum value.
### Step 2: Calculate the x-coordinate where the minimum value occurs
The x-coordinate of the vertex (where the minimum or maximum value occurs) of a quadratic function [tex]\( ax^2 + bx + c \)[/tex] is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
For our function, [tex]\( a = 1 \)[/tex] and [tex]\( b = 4 \)[/tex]:
[tex]\[
x = -\frac{4}{2 \cdot 1} = -\frac{4}{2} = -2
\][/tex]
So, the minimum value occurs at [tex]\( x = -2 \)[/tex].
### Step 3: Calculate the function's minimum value
To find the minimum value of the function, we need to evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = -2 \)[/tex].
Substitute [tex]\( x = -2 \)[/tex] into the function [tex]\( g(x) = x^2 + 4x + 3 \)[/tex]:
[tex]\[
g(-2) = (-2)^2 + 4(-2) + 3 = 4 - 8 + 3 = -1
\][/tex]
So, the minimum value of the function is [tex]\(-1\)[/tex].
### Final Result
- Does the function have a minimum or maximum value?
- Minimum
- Where does the minimum value occur?
[tex]\[
x = -2
\][/tex]
- What is the function's minimum value?
[tex]\[
-1
\][/tex]
