To find the exponential equation that fits the points in the table, we can use the general form of an exponential function:
\[ y = ab^x \]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the growth factor.
The points given in the table are \((-1, 10)\) and \( (2, 2160)\).
Step 1: Use the first point to set up the equation
Using the point \((-1, 10)\):
\[ 10 = ab^{-1} \]
This simplifies to:
\[ 10 = \frac{a}{b} \tag{1} \]
Step 2: Use the second point to set up another equation
Using the point \((2, 2160)\):
\[ 2160 = ab^{2} \] \tag{2}
Step 3: Solve the equations
From equation (1): \[ a = 10b \tag{3} \]
Now, substitute equation (3) into equation (2):
\[ 2160 = (10b)b^{2} \]
This simplifies to:
\[ 2160 = 10b^{3} \]
Now divide both sides by 10:
\[ 216 = b^{3} \]
Now take the cube root of both sides:
\[ b = \sqrt[3]{216} = 6 \]
Step 4: Find \( a \)
Now, substitute \( b = 6 \) back into equation (3) to find \( a \):
\[ a = 10b = 10 \times 6 = 60 \]
Step 5: Write the final exponential equation
Now that we have both \( a \) and \( b \):
\[ y = 60 \cdot 6^x \]
Thus, the exponential equation that contains the points shown in the table is:
\[ \boxed{y = 60 \cdot 6^x} \]
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