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Which table represents a linear function?

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

\begin{tabular}{|l|l|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 0 & 0 \\
\hline 1 & 1 \\
\hline 2 & 4 \\
\hline 3 & 9 \\
\hline
\end{tabular}

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

\begin{tabular}{|c|c|}
\hline[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 0 & 0 \\
\hline 1 & -1 \\
\hline 2 & 4 \\
\hline 3 & -9 \\
\hline
\end{tabular}

Sagot :

GinnyAnsweridn
To determine which table represents a linear function, we can check if the differences between consecutive [tex]\( y \)[/tex]-values are consistent. A table represents a linear function if [tex]\( y \)[/tex] changes by a constant amount as [tex]\( x \)[/tex] increases by 1.

Let's go through each table:

Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 1 \\ \hline 1 & 3 \\ \hline 2 & -3 \\ \hline 3 & -1 \\ \hline \end{array} \][/tex]

Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ 3 - 1 = 2 \][/tex]
[tex]\[ -3 - 3 = -6 \][/tex]
[tex]\[ -1 - (-3) = 2 \][/tex]

The differences are [tex]\( 2, -6, 2 \)[/tex], which are not consistent. Therefore, Table 1 does not represent a linear function.

Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & 1 \\ \hline 2 & 4 \\ \hline 3 & 9 \\ \hline \end{array} \][/tex]

Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ 1 - 0 = 1 \][/tex]
[tex]\[ 4 - 1 = 3 \][/tex]
[tex]\[ 9 - 4 = 5 \][/tex]

The differences are [tex]\( 1, 3, 5 \)[/tex], which are not consistent. Therefore, Table 2 does not represent a linear function.

Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 3 \\ \hline 1 & 1 \\ \hline 2 & -1 \\ \hline 3 & -3 \\ \hline \end{array} \][/tex]

Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ 1 - 3 = -2 \][/tex]
[tex]\[ -1 - 1 = -2 \][/tex]
[tex]\[ -3 - (-1) = -2 \][/tex]

The differences are [tex]\( -2, -2, -2 \)[/tex], which are consistent. Therefore, Table 3 represents a linear function.

Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 0 & 0 \\ \hline 1 & -1 \\ \hline 2 & 4 \\ \hline 3 & -9 \\ \hline \end{array} \][/tex]

Calculate the differences between consecutive [tex]\( y \)[/tex]-values:
[tex]\[ -1 - 0 = -1 \][/tex]
[tex]\[ 4 - (-1) = 5 \][/tex]
[tex]\[ -9 - 4 = -13 \][/tex]

The differences are [tex]\( -1, 5, -13 \)[/tex], which are not consistent. Therefore, Table 4 does not represent a linear function.

Based on our analysis, the table that represents a linear function is:
[tex]\[ \boxed{3} \][/tex]
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