(a) Finding the Rate of Change of $f$

The rate of change of a linear function is the same as the slope of the line it represents. The slope $m$ can be found using the formula:

$m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Using the given values, let's take two points from the table, say $(2, 6)$ and $(0, 3)$:

$m = \frac{6 - 3}{2 - 0} = \frac{3}{2} = 1.5$

So, the rate of change of $f$ is $1.5$.

(b) Expressing $f$ in the Form $f(x) = ax + b$

A linear function can be expressed as $f(x) = ax + b$, where:

• $a$ is the slope (rate of change).
• $b$ is the y-intercept (the value of $f(x)$ when $x = 0$).

From part (a), we found that the slope $a = 1.5$. The y-intercept $b$ is given by $f(0)$, which is $3$.

So, the function $f(x)$ can be written as:

$f(x) = 1.5x + 3$

This is the required linear function.

Summary:

• (a) Rate of change of $f$: $1.5$
• (b) Linear function $f(x) = 1.5x + 3$

Would you like more details or have any questions? Here are 5 related questions you might consider:

1. How do you interpret the rate of change in a real-world context?
2. Can we determine the linear function using any other two points from the table?
3. What does the y-intercept represent in this function?
4. How would the equation change if the table values were scaled by a factor of 2?
5. Can you plot this function on a graph?

Tip: When finding the rate of change, make sure to choose two points that are as far apart as possible for better accuracy in the case of real-world data!