IDNLearn.com provides a comprehensive solution for all your question and answer needs. Join our interactive Q&A platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
To determine which table represents a linear function, we will examine the relationship between the [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values in each table.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 15 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 7 - 3 = 4 \][/tex]
[tex]\[ y_3 - y_2 = 11 - 7 = 4 \][/tex]
[tex]\[ y_4 - y_3 = 15 - 11 = 4 \][/tex]
Since the differences (slopes) are constant (4 each), Table 1 represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 8 \\ \hline 3 & 15 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 8 - 3 = 5 \][/tex]
[tex]\[ y_3 - y_2 = 15 - 8 = 7 \][/tex]
[tex]\[ y_4 - y_3 = 21 - 15 = 6 \][/tex]
The differences (slopes) are not constant (5, 7, 6). Hence, Table 2 does not represent a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 3 \\ \hline 4 & 9 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 9 - 3 = 6 \][/tex]
[tex]\[ y_3 - y_2 = 3 - 9 = -6 \][/tex]
[tex]\[ y_4 - y_3 = 9 - 3 = 6 \][/tex]
The differences (slopes) are not constant (6, -6, 6). Therefore, Table 3 does not represent a linear function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline \end{array} \][/tex]
Let's calculate the difference between the [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 9 - 3 = 6 \][/tex]
Since we only have two points, we can't definitively determine linearity from one difference.
### Conclusion:
After evaluating all the tables, we find that Table 1 is the one where the differences between the [tex]\(y\)[/tex]-values are constant, which means Table 1 represents a linear function.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 7 \\ \hline 3 & 11 \\ \hline 4 & 15 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 7 - 3 = 4 \][/tex]
[tex]\[ y_3 - y_2 = 11 - 7 = 4 \][/tex]
[tex]\[ y_4 - y_3 = 15 - 11 = 4 \][/tex]
Since the differences (slopes) are constant (4 each), Table 1 represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 8 \\ \hline 3 & 15 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 8 - 3 = 5 \][/tex]
[tex]\[ y_3 - y_2 = 15 - 8 = 7 \][/tex]
[tex]\[ y_4 - y_3 = 21 - 15 = 6 \][/tex]
The differences (slopes) are not constant (5, 7, 6). Hence, Table 2 does not represent a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline 3 & 3 \\ \hline 4 & 9 \\ \hline \end{array} \][/tex]
Let's calculate the differences between consecutive [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 9 - 3 = 6 \][/tex]
[tex]\[ y_3 - y_2 = 3 - 9 = -6 \][/tex]
[tex]\[ y_4 - y_3 = 9 - 3 = 6 \][/tex]
The differences (slopes) are not constant (6, -6, 6). Therefore, Table 3 does not represent a linear function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 9 \\ \hline \end{array} \][/tex]
Let's calculate the difference between the [tex]\(y\)[/tex]-values:
[tex]\[ y_2 - y_1 = 9 - 3 = 6 \][/tex]
Since we only have two points, we can't definitively determine linearity from one difference.
### Conclusion:
After evaluating all the tables, we find that Table 1 is the one where the differences between the [tex]\(y\)[/tex]-values are constant, which means Table 1 represents a linear function.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your reliable source for answers. We appreciate your visit and look forward to assisting you again soon.