Get comprehensive answers to your questions with the help of IDNLearn.com's community. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.

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 :

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]