To determine which of the given quadratic functions have the same vertex, let's analyze each function individually.
### Function I: [tex]\( f(x) = (x + 2)^2 + 5 \)[/tex]
Rewriting the function in vertex form [tex]\( f(x) = a(x - h)^2 + k \)[/tex]:
- [tex]\( f(x) = (x + 2)^2 + 5 \)[/tex]
- Comparing, we get [tex]\( a = 1 \)[/tex], [tex]\( h = -2 \)[/tex], and [tex]\( k = 5 \)[/tex].
So, the vertex of [tex]\( f(x) \)[/tex] is [tex]\( (-2, 5) \)[/tex].
### Function II: Given data
[tex]\[
\begin{tabular}{|c|r|r|r|r|r|r|}
\hline
$x$ & -4 & -3 & -2 & -1 & 0 & 1 \\
\hline
$g(x)$ & -3 & 2 & 5 & 5 & 2 & -3 \\
\hline
\end{tabular}
\][/tex]
To find the vertex of this function, we need to determine the maximum (or minimum) point. From the table, we see that the maximum value of [tex]\( g(x) = 5 \)[/tex] occurs at two points: [tex]\( x = -2 \)[/tex] and [tex]\( x = -1 \)[/tex].
For a quadratic function to be symmetric, the vertex lies exactly halfway between these points. Thus, the x-coordinate of the vertex is:
[tex]\[ x = \frac{-2 + (-1)}{2} = \frac{-3}{2} = -1.5 \][/tex]
Substituting [tex]\( x = -1.5 \)[/tex] into the function would confirm a y-value, but we can see directly from the symmetrically placed values around [tex]\( x = -1.5 \)[/tex] that the y-coordinate of the vertex remains at 5.
So, the vertex of [tex]\( g(x) \)[/tex] is [tex]\( (-1.5, 5) \)[/tex].
### Function III: Not enough information
Since no specific details or data points are given for function III, we cannot determine the exact vertex without additional information.
### Comparing Vertices
- Vertex of [tex]\( f(x) \)[/tex] (Function I): [tex]\( (-2, 5) \)[/tex]
- Vertex of [tex]\( g(x) \)[/tex] (Function II): [tex]\( (-1.5, 5) \)[/tex]
- Function III: No vertex data provided
### Conclusion
Given the above analyses, Functions I and II do not have the same vertex. There is no information available for Function III to compare its vertex.
Thus, the correct answer is:
"The quadratic functions do not have the same vertex. Therefore, none of the options 1, 2, 3, or 4 applies."
