A table of values for a linear function f is given.
x	f(x)
0	3 
2	6 
4	9 
6	12 
8	15 
(a)
Find the rate of change of f.
(b)
Express f in the form 
f(x) = ax + b.
f(x) = 
 

### (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!

A table of values for a linear function f is given.
x  f(x)
0  3
2  6
4  9
6  12
8  15
(a) Find the rate of change of f.
(b) Express f in the form f(x) = ax + b.

Learn how to find the rate of change of a linear function using a table of values and express it in the form f(x) = ax + b. This problem uses key concepts in algebra, such as the slope formula and slope-intercept form, suitable for students in Grades 6-8.

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