IDNLearn.com is designed to help you find the answers you need quickly and easily. Find the answers you need quickly and accurately with help from our knowledgeable and dedicated community members.
Sagot :
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.
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.
Your engagement is important to us. Keep sharing your knowledge and experiences. Let's create a learning environment that is both enjoyable and beneficial. Your questions deserve accurate answers. Thank you for visiting IDNLearn.com, and see you again for more solutions.