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Which table represents a linear function?

\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & 3 \\
\hline 2 & 6 \\
\hline 3 & 12 \\
\hline 4 & 24 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & 2 \\
\hline 2 & 5 \\
\hline 3 & 9 \\
\hline 4 & 14 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & -3 \\
\hline 2 & -5 \\
\hline 3 & -7 \\
\hline 4 & -9 \\
\hline
\end{tabular}

\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & -2 \\
\hline 2 & -4 \\
\hline 3 & -2 \\
\hline 4 & 0 \\
\hline
\end{tabular}

Sagot :

GinnyAnswer
To determine which of the given tables represents a linear function, we need to examine the changes in \( y \) for each increment in \( x \). A linear function has a constant rate of change (slope), and this means the difference between consecutive \( y \) values should be the same.

We have four different tables of \( x \) and \( y \) values.

Let's analyze each table:

1. First Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline 3 & 12 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
Differences in \( y \) values:
[tex]\[ 6 - 3 = 3 \\ 12 - 6 = 6 \\ 24 - 12 = 12 \][/tex]
Since the differences are not constant, this does not represent a linear function.

2. Second Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 5 \\ \hline 3 & 9 \\ \hline 4 & 14 \\ \hline \end{array} \][/tex]
Differences in \( y \) values:
[tex]\[ 5 - 2 = 3 \\ 9 - 5 = 4 \\ 14 - 9 = 5 \][/tex]
Since the differences are not constant, this does not represent a linear function.

3. Third Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -3 \\ \hline 2 & -5 \\ \hline 3 & -7 \\ \hline 4 & -9 \\ \hline \end{array} \][/tex]
Differences in \( y \) values:
[tex]\[ -5 - (-3) = -2 \\ -7 - (-5) = -2 \\ -9 - (-7) = -2 \][/tex]
Since the differences are constant (\(-2\)), this represents a linear function.

4. Fourth Table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -2 \\ \hline 2 & -4 \\ \hline 3 & -2 \\ \hline 4 & 0 \\ \hline \end{array} \][/tex]
Differences in \( y \) values:
[tex]\[ -4 - (-2) = -2 \\ -2 - (-4) = 2 \\ 0 - (-2) = 2 \][/tex]
Since the differences are not constant, this does not represent a linear function.

Based on the above analysis, the only table that represents a linear function is the third table:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -3 \\ \hline 2 & -5 \\ \hline 3 & -7 \\ \hline 4 & -9 \\ \hline \end{array} \][/tex]
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