To determine whether the given points are linear or not, we need to explore the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] by calculating the slopes between consecutive points. If the slopes between all pairs of points are the same, the points lie on a straight line, and hence the function is linear. Otherwise, the function is not linear.
Let's list the points provided in the table:
- (2, -2)
- (6, -3)
- (10, -6)
- (14, -8)
Step 1: Compute the differences between consecutive [tex]\( x \)[/tex] values.
For the [tex]\( x \)[/tex] values:
- [tex]\( x_2 - x_1 = 6 - 2 = 4 \)[/tex]
- [tex]\( x_3 - x_2 = 10 - 6 = 4 \)[/tex]
- [tex]\( x_4 - x_3 = 14 - 10 = 4 \)[/tex]
So the differences [tex]\( \Delta x \)[/tex] are:
[tex]\[ [4, 4, 4] \][/tex]
Step 2: Compute the differences between consecutive [tex]\( y \)[/tex] values.
For the [tex]\( y \)[/tex] values:
- [tex]\( y_2 - y_1 = -3 - (-2) = -3 + 2 = -1 \)[/tex]
- [tex]\( y_3 - y_2 = -6 - (-3) = -6 + 3 = -3 \)[/tex]
- [tex]\( y_4 - y_3 = -8 - (-6) = -8 + 6 = -2 \)[/tex]
So the differences [tex]\( \Delta y \)[/tex] are:
[tex]\[ [-1, -3, -2] \][/tex]
Step 3: Compute the slopes between each pair of consecutive points.
The slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
- Slope between (2, -2) and (6, -3):
[tex]\[ \frac{-3 - (-2)}{6 - 2} = \frac{-1}{4} = -0.25 \][/tex]
- Slope between (6, -3) and (10, -6):
[tex]\[ \frac{-6 - (-3)}{10 - 6} = \frac{-3}{4} = -0.75 \][/tex]
- Slope between (10, -6) and (14, -8):
[tex]\[ \frac{-8 - (-6)}{14 - 10} = \frac{-2}{4} = -0.5 \][/tex]
So the slopes are:
[tex]\[ [-0.25, -0.75, -0.5] \][/tex]
Step 4: Determine linearity.
For the points to be linear, all the calculated slopes must be equal. Here we have slopes:
[tex]\[ [-0.25, -0.75, -0.5] \][/tex]
which are not all equal. This difference in slopes indicates that the points do not lie on the same straight line.
Conclusion:
The function described by the given points is not linear.
