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Select the correct answer.

This table represents a quadratic function.
\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & 3 \\
\hline 3 & -3 \\
\hline 5 & -5 \\
\hline 7 & -3 \\
\hline 9 & 3 \\
\hline
\end{tabular}

Which statement about this function is true?

A. The value of [tex]$a$[/tex] is negative, so the vertex is a maximum.

B. The value of [tex]$a$[/tex] is positive, so the vertex is a maximum.

C. The value of [tex]$a$[/tex] is positive, so the vertex is a minimum.

D. The value of [tex]$a$[/tex] is negative, so the vertex is a minimum.

Sagot :

GinnyAnsweridn
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.
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