Question

These tables represent an exponential function. Find the average rate of change for
the interval from x = 8 to x = 9.
x y
0
1
1
2
3
4
5
6
3
9
27
81
243
729
Interval
0 to 1
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
Average rate of change
2
6
54
162
486
]x3
]×3
]×3
]×3
-]×3
0 19,683
0 13,122
0 6561
03 These tables represent an exponential function. Find the average rate of change for
the interval from x = 8 to x = 9.
x y
1
1
2
3
4
5
6
3
9
27
81
243
729
Interval
0 to 1
1 to 2
2 to 3
3 to 4
4 to 5
5 to 6
Average rate of change
2
6
54
162
486
]x3
]×3
]×3
]×3
-]×3
0 19,683
0 13,122
0 6561
03

Answer 1

To find the average rate of change of the exponential function between \( x = 8 \) and \( x = 9 \), we'll first need the corresponding \( y \) values for these \( x \) values, based on the pattern given in the table.

From the earlier parts of the table, we see that the exponential function appears to be of the form \( y = 3^x \). Let's calculate \( y \) for \( x = 8 \) and \( x = 9 \):

1. For \( x = 8 \):
\[
y = 3^8 = 6561
\]

2. For \( x = 9 \):
\[
y = 3^9 = 19683
\]

Now we can calculate the average rate of change over the interval from \( x = 8 \) to \( x = 9 \):

The average rate of change formula is:
\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]
where \( a = 8 \) and \( b = 9 \).

Substituting the values:
\[
\text{Average Rate of Change} = \frac{19683 - 6561}{9 - 8} = \frac{13122}{1} = 13122
\]

Thus, the average rate of change for the interval from \( x = 8 \) to \( x = 9 \) is **13122**.