Discover answers to your most pressing questions at Westonci.ca, the ultimate Q&A platform that connects you with expert solutions. Connect with professionals ready to provide precise answers to your questions on our comprehensive Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To determine which of the given tables represents a linear function, we need to examine whether the differences between consecutive \( y \)-values are consistent, indicating a constant rate of change, or slope.
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & \frac{3}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(\frac{1}{2}, 1, \frac{3}{2}, 2 \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 1 - \frac{1}{2} = \frac{1}{2} \\ \frac{3}{2} - 1 = \frac{1}{2} \\ 2 - \frac{3}{2} = \frac{1}{2} \\ \end{array} \][/tex]
The differences are constant (\(\frac{1}{2}\)), indicating that Table 1 represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{1} \\ \hline 2 & \frac{1}{2} \\ \hline 3 & \frac{1}{3} \\ \hline 4 & \frac{1}{4} \\ \hline \end{array} \][/tex]
The \( y \)-values are \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} \frac{1}{2} - 1 \neq \text{constant} \\ \frac{1}{3} - \frac{1}{2} \neq \text{constant} \\ \frac{1}{4} - \frac{1}{3} \neq \text{constant} \end{array} \][/tex]
The differences are not consistent, indicating that Table 2 does not represent a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 7 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(7, 9, 13, 21\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 9 - 7 = 2 \\ 13 - 9 = 4 \\ 21 - 13 = 8 \end{array} \][/tex]
The differences are not constant, indicating that Table 3 does not represent a linear function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0 \\ \hline 2 & 6 \\ \hline 3 & 16 \\ \hline 4 & 30 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(0, 6, 16, 30\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 6 - 0 = 6 \\ 16 - 6 = 10 \\ 30 - 16 = 14 \end{array} \][/tex]
The differences are not constant, indicating that Table 4 does not represent a linear function.
### Conclusion:
Upon examining all the tables, Table 1 is the only one that has consistent differences between the \( y \)-values. Therefore, Table 1 represents a linear function.
Thus, the table that represents a linear function is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & \frac{3}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
### Table 1:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & \frac{3}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(\frac{1}{2}, 1, \frac{3}{2}, 2 \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 1 - \frac{1}{2} = \frac{1}{2} \\ \frac{3}{2} - 1 = \frac{1}{2} \\ 2 - \frac{3}{2} = \frac{1}{2} \\ \end{array} \][/tex]
The differences are constant (\(\frac{1}{2}\)), indicating that Table 1 represents a linear function.
### Table 2:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{1} \\ \hline 2 & \frac{1}{2} \\ \hline 3 & \frac{1}{3} \\ \hline 4 & \frac{1}{4} \\ \hline \end{array} \][/tex]
The \( y \)-values are \(\frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \frac{1}{4} \).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} \frac{1}{2} - 1 \neq \text{constant} \\ \frac{1}{3} - \frac{1}{2} \neq \text{constant} \\ \frac{1}{4} - \frac{1}{3} \neq \text{constant} \end{array} \][/tex]
The differences are not consistent, indicating that Table 2 does not represent a linear function.
### Table 3:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 7 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 21 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(7, 9, 13, 21\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 9 - 7 = 2 \\ 13 - 9 = 4 \\ 21 - 13 = 8 \end{array} \][/tex]
The differences are not constant, indicating that Table 3 does not represent a linear function.
### Table 4:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 0 \\ \hline 2 & 6 \\ \hline 3 & 16 \\ \hline 4 & 30 \\ \hline \end{array} \][/tex]
The \( y \)-values are \(0, 6, 16, 30\).
Calculate the differences between consecutive \( y \)-values:
[tex]\[ \begin{array}{l} 6 - 0 = 6 \\ 16 - 6 = 10 \\ 30 - 16 = 14 \end{array} \][/tex]
The differences are not constant, indicating that Table 4 does not represent a linear function.
### Conclusion:
Upon examining all the tables, Table 1 is the only one that has consistent differences between the \( y \)-values. Therefore, Table 1 represents a linear function.
Thus, the table that represents a linear function is:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & \frac{1}{2} \\ \hline 2 & 1 \\ \hline 3 & \frac{3}{2} \\ \hline 4 & 2 \\ \hline \end{array} \][/tex]
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
