Question

The table represents some points on the graph of linear function `f`.

x -2 1 5 10
F (x) -224 64 448 928

Which function represents `f`?

F (x) = 32 (3x - 1)
F (x) = - 32 (x - 3)
F (x) = -2 (32x - 3)
F (x) = 16 (2x - 1)

Answer 1

To determine the linear function \( f(x) \) that fits the given points, we can look at the points pairs to find a slope and y-intercept.

Given points are:
- \( (-2, -224) \)
- \( (1, 64) \)
- \( (5, 448) \)
- \( (10, 928) \)

First, let's calculate the slope (m) using two of the points. We'll use \( (1, 64) \) and \( (5, 448) \) for this calculation:

\[
m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} = \frac{448 - 64}{5 - 1} = \frac{384}{4} = 96
\]

Now that we have the slope, we can use one point to find the y-intercept (b). Let's use the point \( (1, 64) \):

\[
f(x) = mx + b \implies 64 = 96(1) + b \implies b = 64 - 96 = -32
\]

Thus, the equation of the line is:

\[
f(x) = 96x - 32
\]

Now, let's identify which of the given functions might represent this linear function. We can rewrite our function in the form near the answers provided:

1. **Check each function to see if any of them matches \( 96x - 32 \)**.

#### Check each option:

1. **For \( F(x) = 32(3x - 1) \)**:
\[
F(x) = 96x - 32
\]

This function matches, so we have that one as a potential answer.

2. **For \( F(x) = -32(x - 3) \)**:
\[
F(x) = -32x + 96
\]
This does not match \( 96x - 32 \).

3. **For \( F(x) = -2(32x - 3) \)**:
\[
F(x) = -64x + 6
\]
This does not match either.

4. **For \( F(x) = 16(2x - 1) \)**:
\[
F(x) = 32x - 16
\]
This does not match.

Thus, the function representing \( f(x) \) is

\[
F(x) = 32(3x - 1).
\]