Question

Find the exponential function that models the data in the table below.
                                                                                                                      
x
negative 4
negative 3
negative 2
negative 1
0
1
2
3
y
StartFraction 8 Over 81 EndFraction
StartFraction 8 Over 27 EndFraction
eight ninths
eight thirds
8
24
72
216
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Part 1
What is the exponential regression of the​ data?
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Answer 1

To find the exponential function that models the given data, we typically use the form:

\[
y = a \cdot b^x
\]

where \(a\) is the initial value (the value of \(y\) when \(x = 0\)) and \(b\) is the base that describes the growth rate.

The data provided is:

\[
\begin{aligned}
& x: -4, -3, -2, -1, 0, 1, 2, 3 \\
& y: \frac{8}{81}, \frac{8}{27}, \frac{8}{9}, \frac{8}{3}, 8, 24, 72, 216
\end{aligned}
\]

1. **Finding the initial value \(a\)**: From the data, when \(x = 0\), \(y = 8\). Therefore, \(a = 8\).

2. **Finding the growth factor \(b\)**: We can calculate \(b\) by using two points (`(0, 8)` and `(1, 24)`):

Using the point \( (0, 8) \) to \( (1, 24) \):
\[
y(1) = a \cdot b^1 = 8b = 24
\]

Solving for \(b\):
\[
b = \frac{24}{8} = 3
\]

3. **Putting it all together**: Now we can write the exponential function:

\[
y = 8 \cdot 3^x
\]

Thus, the exponential regression of the data is:

\[
\boxed{y = 8 \cdot 3^x}
\]

You can also verify the correctness of this model by plugging in other values of \(x\) and checking the outputs \(y\) match the values in the table.