Answer the questions below about the quadratic function:

[tex]\[ g(x) = x^2 + 4x + 3 \][/tex]

1. Does the function have a minimum or maximum value?
- Minimum
- Maximum

2. Where does the minimum or maximum value occur?
[tex]\[ x = \square \][/tex]

3. What is the function's minimum or maximum value?
[tex]\[ \square \][/tex]



Answer :

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]