Sure, let's solve this step-by-step.
Given the quadratic function [tex]\( y = x^2 + 8x + 3 \)[/tex], we will rewrite it in vertex form by completing the square.
### Step 1: Start with the original quadratic function
[tex]\[ y = x^2 + 8x + 3 \][/tex]
### Step 2: Factor out the coefficient of [tex]\(x^2\)[/tex]
In this case, the coefficient is 1, so we don't need to factor anything out.
### Step 3: Complete the square
We need to add and subtract a certain value to complete the square inside the expression. This value is found using
[tex]\[ \left(\frac{b}{2}\right)^2 \][/tex]
where [tex]\( b \)[/tex] is the coefficient of [tex]\( x \)[/tex]. Here, [tex]\( b = 8 \)[/tex], so:
[tex]\[ \left(\frac{8}{2}\right)^2 = 16 \][/tex]
We add and subtract 16 inside the equation:
[tex]\[ y = x^2 + 8x + 16 - 16 + 3 \][/tex]
### Step 4: Rewrite as a perfect square trinomial and simplify
Now, we can rewrite the trinomial [tex]\( x^2 + 8x + 16 \)[/tex] as a square of a binomial:
[tex]\[ y = (x + 4)^2 - 16 + 3 \][/tex]
[tex]\[ y = (x + 4)^2 - 13 \][/tex]
### Step 5: Write in vertex form
The vertex form of a quadratic function is expressed as:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola. Here, we have:
[tex]\[ y = (x + 4)^2 - 13 \][/tex]
which means [tex]\( (x + 4)^2 = (x - (-4))^2 \)[/tex]. This gives us:
[tex]\[ h = -4 \][/tex]
and
[tex]\[ k = -13 \][/tex]
Therefore, the minimum value of [tex]\( y \)[/tex] occurs at the vertex, which is:
[tex]\[ y = -13 \][/tex]
### Conclusion
The minimum [tex]\( y \)[/tex]-value of the function is:
[tex]\[ -13 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{-13} \][/tex]
So, the minimum value is [tex]\( -13 \)[/tex], which corresponds to option C.
