To find the linear function that best fits the given data points, we first recognize that we are looking for a linear equation of the form:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
Given Data:
- \( x: \) 1, 3, 5, 7, 9
- \( y: \) 5, 8, 11, 14, 17
Step 1: Calculate the slope (m)
The slope \( m \) of the line can be calculated using the formula:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
where \( N \) is the number of data points.
Calculating necessary sums:
- \( N = 5 \) (number of data points)
- \( \sum x = 1 + 3 + 5 + 7 + 9 = 25 \)
- \( \sum y = 5 + 8 + 11 + 14 + 17 = 55 \)
- \( \sum xy = (1)(5) + (3)(8) + (5)(11) + (7)(14) + (9)(17) = 5 + 24 + 55 + 98 + 153 = 335 \)
- \( \sum x^2 = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \)
Step 2: Plug the values into the slope formula
Now, substituting into the formula for \( m \):
\[ m = \frac{5(335) - (25)(55)}{5(165) - (25)^2} \]
Calculating the components:
- \( 5(335) = 1675 \)
- \( (25)(55) = 1375 \)
- \( 5(165) = 825 \)
- \( (25)^2 = 625 \)
Now substituting:
\[ m = \frac{1675 - 1375}{825 - 625} = \frac{300}{200} = 1.5 \]
Step 3: Calculate the y-intercept (b)
The y-intercept \( b \) can be found using the formula:
\[ b = \frac{\sum y - m \sum x}{N} \]
Substituting the known values:
\[ b = \frac{55 - 1.5(25)}{5} \]
Calculating:
- \( 1.5(25) = 37.5 \)
So,
\[ b = \frac{55 - 37.5}{5} = \frac{17.5}{5} = 3.5 \]
Conclusion
The best-fit linear function for the given data points is:
\[ y = 1.5x + 3.5 \]