To determine the interval over which the function [tex]\( k \)[/tex] is increasing, we need to analyze the nature of the quadratic function [tex]\( k(x) = ax^2 + bx + c \)[/tex] given by the ordered pairs:
[tex]\[
\begin{array}{c|cccccc}
x & -1 & 0 & 1 & 2 & 3 & 4 \\
\hline
k(x) & 5 & 8 & 9 & 8 & 5 & 0 \\
\end{array}
\][/tex]
From these points, we can see that the function [tex]\( k(x) \)[/tex] has the general form [tex]\( k(x) = ax^2 + bx + c \)[/tex].
To understand when the quadratic function is increasing, we'll examine its vertex. The vertex of a quadratic function [tex]\( k(x) = ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex]. This vertex represents a maximum or minimum point of the parabola.
Given the form of the quadratic equation and observing the pattern of the values from the table, we can deduce the following:
- The function [tex]\( k(x) \)[/tex] reaches its maximum at [tex]\( x = 1 \)[/tex].
- Quadratic functions with a maximum vertex are increasing to the right of the vertex (i.e., for [tex]\( x > 1 \)[/tex]).
Since the function is symmetric around its vertex and increasing to the right of this point, it will be increasing for all [tex]\( x > 1 \)[/tex].
Thus, the interval over which the function [tex]\( k \)[/tex] is increasing is:
[tex]\[ \boxed{D. (-\infty, 9)} \][/tex]
