IDNLearn.com is your go-to resource for finding answers to any question you have. Get accurate and comprehensive answers to your questions from our community of knowledgeable professionals.
Sagot :
To determine which table represents a linear function, we need to check if the relation between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] follows a constant rate of change, known as the slope. For a function to be linear, the differences between consecutive [tex]\(x\)[/tex] values and their corresponding [tex]\(y\)[/tex] values should result in the same slope when calculated.
Let's examine each table in detail:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline 3 & 12 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = 3, 6, 12\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are 3, 6, and 12)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{3}{1}, \frac{6}{1}, \frac{12}{1} \][/tex]
The slopes are 3, 6, and 12, which are not consistent. Thus, Table 1 does not represent a linear function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 5 \\ \hline 3 & 9 \\ \hline 4 & 14 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = 3, 4, 5\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are 3, 4, and 5)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{3}{1}, \frac{4}{1}, \frac{5}{1} \][/tex]
The slopes are 3, 4, and 5, which are not consistent. Thus, Table 2 does not represent a linear function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -3 \\ \hline 2 & -5 \\ \hline 3 & -7 \\ \hline 4 & -9 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = -2, -2, -2\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are all -2)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{-2}{1}, \frac{-2}{1}, \frac{-2}{1} \][/tex]
The slopes are all -2, which are consistent. Therefore, Table 3 represents a linear function.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -2 \\ \hline 2 & -4 \\ \hline 3 & -2 \\ \hline 4 & 0 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = -2, 2, 2\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are -2, 2, and 2)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{-2}{1}, \frac{2}{1}, \frac{2}{1} \][/tex]
The slopes are -2, 2, and 2, which are not consistent. Thus, Table 4 does not represent a linear function.
### Conclusion
Only Table 3 represents a linear function.
Let's examine each table in detail:
### Table 1
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 3 \\ \hline 2 & 6 \\ \hline 3 & 12 \\ \hline 4 & 24 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = 3, 6, 12\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are 3, 6, and 12)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{3}{1}, \frac{6}{1}, \frac{12}{1} \][/tex]
The slopes are 3, 6, and 12, which are not consistent. Thus, Table 1 does not represent a linear function.
### Table 2
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 2 \\ \hline 2 & 5 \\ \hline 3 & 9 \\ \hline 4 & 14 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = 3, 4, 5\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are 3, 4, and 5)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{3}{1}, \frac{4}{1}, \frac{5}{1} \][/tex]
The slopes are 3, 4, and 5, which are not consistent. Thus, Table 2 does not represent a linear function.
### Table 3
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -3 \\ \hline 2 & -5 \\ \hline 3 & -7 \\ \hline 4 & -9 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = -2, -2, -2\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are all -2)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{-2}{1}, \frac{-2}{1}, \frac{-2}{1} \][/tex]
The slopes are all -2, which are consistent. Therefore, Table 3 represents a linear function.
### Table 4
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & -2 \\ \hline 2 & -4 \\ \hline 3 & -2 \\ \hline 4 & 0 \\ \hline \end{array} \][/tex]
Calculate the differences in [tex]\(x\)[/tex] and [tex]\(y\)[/tex] values:
- [tex]\(\Delta x = 1, 1, 1\)[/tex] (differences between consecutive [tex]\(x\)[/tex] values are all 1)
- [tex]\(\Delta y = -2, 2, 2\)[/tex] (differences between consecutive [tex]\(y\)[/tex] values are -2, 2, and 2)
Calculate the slopes:
[tex]\[ \frac{\Delta y}{\Delta x} = \frac{-2}{1}, \frac{2}{1}, \frac{2}{1} \][/tex]
The slopes are -2, 2, and 2, which are not consistent. Thus, Table 4 does not represent a linear function.
### Conclusion
Only Table 3 represents a linear function.
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.
