To find the linear function that best fits the provided data points, we can use the method of least squares to determine the slope (m) and y-intercept (b) of the linear equation in the form \( y = mx + b \).
Given the data points:
\[ \begin{align*} x: & \quad 1, 3, 5, 7, 9 \ y: & \quad 1, 2, 3, 4, 5 \ \end{align*} \]
Step 1: Calculate the means of \( x \) and \( y \)
\[ \bar{x} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5 \]
\[ \bar{y} = \frac{1 + 2 + 3 + 4 + 5}{5} = \frac{15}{5} = 3 \]
Step 2: Calculate the slope (m)
The formula for the slope \( m \) is:
\[ m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \]
Next, we calculate each component:
\[ \begin{align*} \sum (x_i - \bar{x})(y_i - \bar{y}) & = (1 - 5)(1 - 3) + (3 - 5)(2 - 3) + (5 - 5)(3 - 3) + (7 - 5)(4 - 3) + (9 - 5)(5 - 3) \ & = (-4)(-2) + (-2)(-1) + (0)(0) + (2)(1) + (4)(2) \ & = 8 + 2 + 0 + 2 + 8 = 20 \end{align*} \]
\[ \sum (x_i - \bar{x})^2 = (1 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (7 - 5)^2 + (9 - 5)^2 \ = 16 + 4 + 0 + 4 + 16 = 40 \]
Now we can calculate \( m \):
\[ m = \frac{20}{40} = 0.5 \]
Step 3: Calculate the y-intercept (b)
The formula for the y-intercept \( b \) is:
\[ b = \bar{y} - m\bar{x} \]
Substituting the values we have:
\[ b = 3 - 0.5 \cdot 5 = 3 - 2.5 = 0.5 \]
Finally, the equation of the linear function is:
\[ y = 0.5x + 0.5 \]
Part 1 Answer:
The linear function that best fits the data is:
\[ y = 0.5x + 0.5 \]