To write the exponential equation containing the points from the table, we have two points:
- \((-1, 10)\)
- \((2, 2160)\)
An exponential function can be written in the form:
\[ y = ab^x \]
where \(a\) is the initial value when \(x=0\), and \(b\) is the growth factor.
Using the points given:
- For the point \((-1, 10)\):
\[ 10 = ab^{-1} \implies 10 = \frac{a}{b} \quad (1) \]
- For the point \((2, 2160)\):
\[ 2160 = ab^2 \quad (2) \]
Now we have two equations. We can solve for \(a\) and \(b\).
From equation (1):
\[ a = 10b \quad (3) \]
Substituting equation (3) into equation (2):
\[ 2160 = (10b)b^2 \] \[ 2160 = 10b^3 \] \[ b^3 = \frac{2160}{10} = 216 \] \[ b = \sqrt[3]{216} = 6 \]
Now substitute \(b = 6\) back into equation (3) to find \(a\):
\[ a = 10(6) = 60 \]
Putting this all together, the exponential equation is:
\[ y = 60 \cdot 6^x \]
Thus, the final exponential equation is:
\[ \boxed{y = 60 \cdot 6^x} \]
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