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Sagot :
To determine the equation of the line that best fits the given points, we need to find the slope and the intercept of the line. The points given are:
[tex]\[ (-2, 12), (-1, 7), (0, 2), (1, -3), (2, -8) \][/tex]
Follow these steps to find the equation of the line:
1. List the x- and y-values:
[tex]\[ \begin{aligned} &x: -2, -1, 0, 1, 2 \\ &y: 12, 7, 2, -3, -8 \\ \end{aligned} \][/tex]
2. Calculate the slope (m) and intercept (b) using the least squares method:
The general form of the linear equation is:
[tex]\[ y = mx + b \][/tex]
We need to calculate the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex]. The formulas for [tex]\( m \)[/tex] and [tex]\( b \)[/tex] when we have a set of points [tex]\((x_i, y_i)\)[/tex] are:
[tex]\[ m = \frac{n(\sum x_i y_i) - (\sum x_i)(\sum y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \][/tex]
[tex]\[ b = \frac{(\sum y_i)(\sum x_i^2) - (\sum x_i)(\sum x_i y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \][/tex]
Here, [tex]\( n \)[/tex] is the number of points, which is 5 in this case.
Given our data, we have:
[tex]\[ \begin{aligned} & \sum x_i = -2 + (-1) + 0 + 1 + 2 = 0 \\ & \sum y_i = 12 + 7 + 2 - 3 - 8 = 10 \\ & \sum x_i y_i = (-2)(12) + (-1)(7) + (0)(2) + (1)(-3) + (2)(-8) = -24 - 7 + 0 - 3 - 16 = -50 \\ & \sum x_i^2 = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10 \\ \end{aligned} \][/tex]
Now plug these sums into the formulas for [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ m = \frac{5(-50) - (0)(10)}{5(10) - (0)^2} = \frac{-250}{50} = -5 \][/tex]
[tex]\[ b = \frac{(10)(10) - (0)(-50)}{5(10) - (0)^2} = \frac{100}{50} = 2 \][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex] and the intercept [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
3. Write the equation of the line:
[tex]\[ y = -5x + 2 \][/tex]
4. Match the line equation with the options given:
D. [tex]\( y=-5x+2 \)[/tex]
Hence, the equation of the line represented by the table of points is:
[tex]\[ \boxed{D. \, y = -5x + 2} \][/tex]
[tex]\[ (-2, 12), (-1, 7), (0, 2), (1, -3), (2, -8) \][/tex]
Follow these steps to find the equation of the line:
1. List the x- and y-values:
[tex]\[ \begin{aligned} &x: -2, -1, 0, 1, 2 \\ &y: 12, 7, 2, -3, -8 \\ \end{aligned} \][/tex]
2. Calculate the slope (m) and intercept (b) using the least squares method:
The general form of the linear equation is:
[tex]\[ y = mx + b \][/tex]
We need to calculate the slope [tex]\( m \)[/tex] and the intercept [tex]\( b \)[/tex]. The formulas for [tex]\( m \)[/tex] and [tex]\( b \)[/tex] when we have a set of points [tex]\((x_i, y_i)\)[/tex] are:
[tex]\[ m = \frac{n(\sum x_i y_i) - (\sum x_i)(\sum y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \][/tex]
[tex]\[ b = \frac{(\sum y_i)(\sum x_i^2) - (\sum x_i)(\sum x_i y_i)}{n(\sum x_i^2) - (\sum x_i)^2} \][/tex]
Here, [tex]\( n \)[/tex] is the number of points, which is 5 in this case.
Given our data, we have:
[tex]\[ \begin{aligned} & \sum x_i = -2 + (-1) + 0 + 1 + 2 = 0 \\ & \sum y_i = 12 + 7 + 2 - 3 - 8 = 10 \\ & \sum x_i y_i = (-2)(12) + (-1)(7) + (0)(2) + (1)(-3) + (2)(-8) = -24 - 7 + 0 - 3 - 16 = -50 \\ & \sum x_i^2 = (-2)^2 + (-1)^2 + 0^2 + 1^2 + 2^2 = 4 + 1 + 0 + 1 + 4 = 10 \\ \end{aligned} \][/tex]
Now plug these sums into the formulas for [tex]\( m \)[/tex] and [tex]\( b \)[/tex]:
[tex]\[ m = \frac{5(-50) - (0)(10)}{5(10) - (0)^2} = \frac{-250}{50} = -5 \][/tex]
[tex]\[ b = \frac{(10)(10) - (0)(-50)}{5(10) - (0)^2} = \frac{100}{50} = 2 \][/tex]
Thus, the slope [tex]\( m \)[/tex] is [tex]\(-5\)[/tex] and the intercept [tex]\( b \)[/tex] is [tex]\(2\)[/tex].
3. Write the equation of the line:
[tex]\[ y = -5x + 2 \][/tex]
4. Match the line equation with the options given:
D. [tex]\( y=-5x+2 \)[/tex]
Hence, the equation of the line represented by the table of points is:
[tex]\[ \boxed{D. \, y = -5x + 2} \][/tex]
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