Given the table for f(x) and g(x) below, find the following:


x f(x) g(x)
0 7 6
1 4 1
2 6 2
3 3 7
4 9 3
5 8 5
6 0 9
7 1 0
8 5 4
9 2 8

a) Average rate of change of g(x) from x=6 to x=7 is =


b) Average rate of change of f(x) from x=2 to x=6 is =

To find the average rate of change of a function over a given interval, we use the formula:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

where \( f(a) \) and \( f(b) \) are the function values at the endpoints of the interval \([a, b]\).

### a) Average rate of change of \( g(x) \) from \( x = 6 \) to \( x = 7 \):

1. From the table:
   - \( g(6) = 9 \)
   - \( g(7) = 0 \)

2. Apply the formula:
\[
\text{Average Rate of Change of } g(x) = \frac{g(7) - g(6)}{7 - 6} = \frac{0 - 9}{1} = \frac{-9}{1} = -9
\]

So, the average rate of change of \( g(x) \) from \( x = 6 \) to \( x = 7 \) is \(-9\).

### b) Average rate of change of \( f(x) \) from \( x = 2 \) to \( x = 6 \):

1. From the table:
   - \( f(2) = 6 \)
   - \( f(6) = 0 \)

2. Apply the formula:
\[
\text{Average Rate of Change of } f(x) = \frac{f(6) - f(2)}{6 - 2} = \frac{0 - 6}{4} = \frac{-6}{4} = -1.5
\]

So, the average rate of change of \( f(x) \) from \( x = 2 \) to \( x = 6 \) is \(-1.5\).

Would you like more details on these calculations, or do you have any further questions?

Here are 5 related questions you might find useful:
1. What is the significance of the average rate of change in a function?
2. How does the average rate of change relate to the slope of a line?
3. Can you explain how the rate of change of a linear function differs from that of a non-linear function?
4. How would you calculate the average rate of change if given a continuous function instead of a table of values?
5. How can you determine if a function is increasing or decreasing based on the average rate of change?

**Tip:** The average rate of change between two points on a function is the slope of the line that connects those points. This concept is crucial when analyzing linearity and trends in data.

Given the table for f(x) and g(x) below, find the following:

x    f(x)  g(x)
0    7     6
1    4     1
2    6     2
3    3     7
4    9     3
5    8     5
6    0     9
7    1     0
8    5     4
9    2     8

a) Average rate of change of g(x) from x = 6 to x = 7 is =

b) Average rate of change of f(x) from x = 2 to x = 6 is =

Learn how to calculate the average rate of change for functions f(x) and g(x) using a table of values. This example covers key algebraic concepts, including calculating rates of change over intervals. Suitable for students in Grades 8-10.

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