Question

The table represents some points on the graph of linear function f. x: -3, 2, 5, 11. f(x): -130. 0. 78. 234. which function represents f.

Answer 1

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:
\[
f(x) = 26x - 52
\]