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Sagot :
To determine which table represents a linear function, we need to verify if there is a constant rate of change (slope) between the [tex]\( y \)[/tex]-values for each corresponding [tex]\( x \)[/tex]-value. A linear function will have a constant slope across all points.
Let's examine each table:
### Table A:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 0 & 2 & 5 & 8 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 2 - 0 = 2 \\ \Delta y_2 = 5 - 2 = 3 \\ \Delta y_3 = 8 - 5 = 3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
### Table B:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 10 & 9 & 7 & 4 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 9 - 10 = -1 \\ \Delta y_2 = 7 - 9 = -2 \\ \Delta y_3 = 4 - 7 = -3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
### Table C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 11 & 8 & 5 & 2 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 8 - 11 = -3 \\ \Delta y_2 = 5 - 8 = -3 \\ \Delta y_3 = 2 - 5 = -3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are consistent, this table does represent a linear function.
### Table D:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 8 & 6 & 7 & 5 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 6 - 8 = -2 \\ \Delta y_2 = 7 - 6 = 1 \\ \Delta y_3 = 5 - 7 = -2 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
Thus, the table that represents a linear function is:
### Table C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 11 & 8 & 5 & 2 \\ \hline \end{array} \][/tex]
Let's examine each table:
### Table A:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 0 & 2 & 5 & 8 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 2 - 0 = 2 \\ \Delta y_2 = 5 - 2 = 3 \\ \Delta y_3 = 8 - 5 = 3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
### Table B:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 10 & 9 & 7 & 4 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 9 - 10 = -1 \\ \Delta y_2 = 7 - 9 = -2 \\ \Delta y_3 = 4 - 7 = -3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
### Table C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 11 & 8 & 5 & 2 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 8 - 11 = -3 \\ \Delta y_2 = 5 - 8 = -3 \\ \Delta y_3 = 2 - 5 = -3 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are consistent, this table does represent a linear function.
### Table D:
[tex]\[ \begin{array}{|l|l|l|l|l|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 8 & 6 & 7 & 5 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\( y \)[/tex]-values:
[tex]\[ \Delta y_1 = 6 - 8 = -2 \\ \Delta y_2 = 7 - 6 = 1 \\ \Delta y_3 = 5 - 7 = -2 \][/tex]
Since the differences in [tex]\( y \)[/tex]-values ([tex]\(\Delta y\)[/tex]) are not consistent, this table does not represent a linear function.
Thus, the table that represents a linear function is:
### Table C:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline x & 0 & 1 & 2 & 3 \\ \hline y & 11 & 8 & 5 & 2 \\ \hline \end{array} \][/tex]
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