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:
\[ \begin{aligned} & x: -4, -3, -2, -1, 0, 1, 2, 3 \ & y: \frac{8}{81}, \frac{8}{27}, \frac{8}{9}, \frac{8}{3}, 8, 24, 72, 216 \end{aligned} \]
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Finding the initial value \(a\): From the data, when \(x = 0\), \(y = 8\). Therefore, \(a = 8\).
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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 \(b\): \[ b = \frac{24}{8} = 3 \]
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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.
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