To determine which function represents the linear function \( f \) based on the given points, we can use any two points from the table to calculate the slope of the linear function and then identify the function that matches the calculated slope.
The given points are:
- \( (-3, -130) \)
- \( (2, 0) \)
- \( (5, 78) \)
- \( (11, 234) \)
We'll calculate the slope \( m \) using the formula: \[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]
Let's use the points \( (2, 0) \) and \( (5, 78) \):
- \( x_1 = 2, f(x_1) = 0 \)
- \( x_2 = 5, f(x_2) = 78 \)
Calculating the slope: \[ m = \frac{78 - 0}{5 - 2} = \frac{78}{3} = 26 \]
Now that we have the slope (\( m = 26 \)), we can use one of the points to find the y-intercept \( b \) using the equation of the line \( f(x) = mx + b \). We can use the point \( (2, 0) \): \[ 0 = 26(2) + b \] \[ 0 = 52 + b \implies b = -52 \]
Thus, the function can be expressed as: \[ f(x) = 26x - 52 \]
Now it's time to see if any of the answer choices match this function. Let's rewrite the options: A. \( f(x) = 26 (x - 2) \) expands to \( 26x - 52 \) (which matches!) B. \( f(x) = -26 (2x - 1) \) expands to \( -52x + 26 \) C. \( f(x) = 13 (x - 2) \) expands to \( 13x - 26 \) D. \( f(x) = -2 (26x - 1) \) expands to \( -52x + 2 \)
The only matching option is: A. \( f(x) = 26(x - 2) \).