To find the linear function that best fits the given data points, we can use the method of linear regression. The data points provided are:
\[ \begin{align*} (1, 2) \ (3, 3) \ (5, 4) \ (7, 5) \ (9, 6) \ \end{align*} \]
We want to find the equation of the line in the form:
\[ y = mx + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
Step 1: Calculate the slope \( m \)
The formula for the slope \( m \) is:
\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]
Where:
- \( N \) is the number of data points
- \( x \) is the independent variable and \( y \) is the dependent variable
Let's calculate the necessary components:
\[ \begin{align*} \sum x & = 1 + 3 + 5 + 7 + 9 = 25 \ \sum y & = 2 + 3 + 4 + 5 + 6 = 20 \ \sum xy & = (1 \cdot 2) + (3 \cdot 3) + (5 \cdot 4) + (7 \cdot 5) + (9 \cdot 6) = 2 + 9 + 20 + 35 + 54 = 120 \ \sum x^2 & = 1^2 + 3^2 + 5^2 + 7^2 + 9^2 = 1 + 9 + 25 + 49 + 81 = 165 \ \end{align*} \]
Now, plug everything into the slope formula:
\[ m = \frac{5(120) - (25)(20)}{5(165) - (25)^2} = \frac{600 - 500}{825 - 625} = \frac{100}{200} = 0.5 \]
Step 2: Calculate the y-intercept \( b \)
The formula for the y-intercept \( b \) is:
\[ b = \frac{\sum y - m(\sum x)}{N} \]
Plugging in the values we calculated:
\[ b = \frac{20 - 0.5(25)}{5} = \frac{20 - 12.5}{5} = \frac{7.5}{5} = 1.5 \]
Step 3: Write the linear function
Now, we can formulate the linear function:
\[ y = 0.5x + 1.5 \]
Final Answer
Thus, the linear function that best fits the data is:
\[ y = 0.5x + 1.5 \]