To determine whether the given values could represent a linear function $f(x) = mx + b$, let's follow these steps:

Step 1: Check for a constant rate of change (slope)

For a function to be linear, the slope between any two points must be the same. The slope formula between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

1. Between $(0, -4)$ and $(1, -2)$: $m = \frac{-2 - (-4)}{1 - 0} = \frac{-2 + 4}{1} = \frac{2}{1} = 2$

2. Between $(1, -2)$ and $(2, 0)$: $m = \frac{0 - (-2)}{2 - 1} = \frac{0 + 2}{1} = \frac{2}{1} = 2$

3. Between $(2, 0)$ and $(3, 2)$: $m = \frac{2 - 0}{3 - 2} = \frac{2}{1} = 2$

The slope is constant ($m = 2$) between all points. Therefore, the function is linear.

Step 2: Write the equation $f(x) = mx + b$

We know the slope $m = 2$. Now, to find $b$ (the y-intercept), substitute $x = 0$ and $f(x) = -4$ into the equation:

$f(x) = 2x + b$

$-4 = 2(0) + b$

$b = -4$

Thus, the equation of the linear function is:

$f(x) = 2x - 4$

Final Answer:

The values could represent the linear function $f(x) = 2x - 4$.

Follow-up Questions:

1. How do you determine whether a function is linear using a table of values?
2. What is the significance of the slope in a linear equation?
3. How do you find the y-intercept when the slope and a point are known?
4. Could the function have a different slope and still pass through these points?
5. How do you graph a linear function given its equation?

Tip: When analyzing tables, always calculate the change in $f(x)$ (y-values) relative to the change in $x$ to check for linearity.