To determine which statement about the quadratic function is true, let's analyze the table:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 3 \\
\hline
3 & -3 \\
\hline
5 & -5 \\
\hline
7 & -3 \\
\hline
9 & 3 \\
\hline
\end{array}
\][/tex]
Quadratic functions [tex]\[y = ax^2 + bx + c\][/tex] typically form a parabola. To extract meaningful information:
1. Look for symmetry in the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values. Given the data points [tex]\((x, y) = (1, 3), (3, -3), (5, -5), (7, -3), (9, 3)\)[/tex], notice there's symmetry around [tex]\((5, -5)\)[/tex]. This suggests that [tex]\((5, -5)\)[/tex] is the vertex of the parabola.
2. The vertex is the highest or lowest point of the parabola, and can determine the sign of [tex]\(a\)[/tex]:
- If the parabola opens upwards, [tex]\((5, -5)\)[/tex] would be the lowest point (minimum), so [tex]\(a > 0\)[/tex].
- If the parabola opens downwards, [tex]\((5, -5)\)[/tex] would be the highest point (maximum), so [tex]\(a < 0\)[/tex].
3. Evaluate the values around the vertex:
- For [tex]\(x = 1\)[/tex], [tex]\(y = 3\)[/tex]
- For [tex]\(x = 3\)[/tex], [tex]\(y = -3\)[/tex]
- For [tex]\(x = 7\)[/tex], [tex]\(y = -3\)[/tex]
- For [tex]\(x = 9\)[/tex], [tex]\(y = 3\)[/tex]
For [tex]\(x\)[/tex] values farther from [tex]\(5\)[/tex], [tex]\(y\)[/tex] values increase, indicating the parabola opens downwards. Hence, [tex]\(a\)[/tex] must be negative.
Thus, the vertex [tex]\((5, -5)\)[/tex] is the maximum point because the parabola opens downwards.
So the correct statement is:
A. The value of [tex]\(a\)[/tex] is negative, so the vertex is a maximum.
