Answer :
First, let's extract the given data points from the table:
[tex]\[ \begin{aligned} &(3, 10), \\ &(6, 13), \\ &(9, 16), \\ &(12, 19). \end{aligned} \][/tex]
We are to determine which linear function (segment of the data) has the steepest slope.
To do so, we need to calculate the slope of each segment between consecutive points. The slope \( m \) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
### Calculating the Slopes:
1. Between \((3, 10)\) and \((6, 13)\):
[tex]\[ m_1 = \frac{13 - 10}{6 - 3} = \frac{3}{3} = 1.0 \][/tex]
2. Between \((6, 13)\) and \((9, 16)\):
[tex]\[ m_2 = \frac{16 - 13}{9 - 6} = \frac{3}{3} = 1.0 \][/tex]
3. Between \((9, 16)\) and \((12, 19)\):
[tex]\[ m_3 = \frac{19 - 16}{12 - 9} = \frac{3}{3} = 1.0 \][/tex]
### Analysis:
From these calculations, we see that the slopes of all the segments are \(1.0\).
### Conclusion:
The slopes of all the linear functions (segments between the consecutive points) are equal, each being \(1.0\). Therefore, there is no steepest slope as all slopes are the same.
Thus, the segments all have a slope of 1.0 and none is steeper than the others.
[tex]\[ \begin{aligned} &(3, 10), \\ &(6, 13), \\ &(9, 16), \\ &(12, 19). \end{aligned} \][/tex]
We are to determine which linear function (segment of the data) has the steepest slope.
To do so, we need to calculate the slope of each segment between consecutive points. The slope \( m \) between any two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
### Calculating the Slopes:
1. Between \((3, 10)\) and \((6, 13)\):
[tex]\[ m_1 = \frac{13 - 10}{6 - 3} = \frac{3}{3} = 1.0 \][/tex]
2. Between \((6, 13)\) and \((9, 16)\):
[tex]\[ m_2 = \frac{16 - 13}{9 - 6} = \frac{3}{3} = 1.0 \][/tex]
3. Between \((9, 16)\) and \((12, 19)\):
[tex]\[ m_3 = \frac{19 - 16}{12 - 9} = \frac{3}{3} = 1.0 \][/tex]
### Analysis:
From these calculations, we see that the slopes of all the segments are \(1.0\).
### Conclusion:
The slopes of all the linear functions (segments between the consecutive points) are equal, each being \(1.0\). Therefore, there is no steepest slope as all slopes are the same.
Thus, the segments all have a slope of 1.0 and none is steeper than the others.
