To determine the linear function \( f(x) = mx + b \), we can use the points provided to find the slope \( m \) and the y-intercept \( b \).
The given points are:
- \( (-3, -130) \)
- \( (2, 0) \)
- \( (5, 78) \)
- \( (11, 234) \)
Step 1: Calculate the slope (m)
Using two points, we can calculate the slope \( m \) as follows: \[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Using points \( (2, 0) \) and \( (5, 78) \): \[ m = \frac{78 - 0}{5 - 2} = \frac{78}{3} = 26 \]
Step 2: Finding the y-intercept (b)
Now that we have the slope \( m = 26 \), we can use one of the points to solve for \( b \). Using the point \( (2, 0) \): \[ f(x) = mx + b \implies 0 = 26(2) + b \implies 0 = 52 + b \implies b = -52 \]
Step 3: Write the linear function
Now we can write the linear function: \[ f(x) = 26x - 52 \]
Conclusion
The function that represents \( f \) is: \[ f(x) = 26x - 52 \]