To find the exponential equation that fits the points given in the table, we'll assume that the equation has the form:
\( y = ab^x \)
where \( a \) is the initial value (the value of \( y \) when \( x = 0 \)), and \( b \) is the base of the exponential function.
We have two points from the table:
- \( (-1, 10) \)
- \( (2, 2160) \)
Now we'll use these points to set up a system of equations.
Step 1: Using the point \( (-1, 10) \)
Substituting into the equation:
\[ 10 = ab^{-1} \] This can be rearranged to: \[ 10 = \frac{a}{b} \quad (1) \]
Step 2: Using the point \( (2, 2160) \)
Substituting into the equation:
\[ 2160 = ab^2 \quad (2) \]
Step 3: Substitute Equation (1) into Equation (2)
We can express \( a \) in terms of \( b \) using Equation (1):
\[ a = 10b \quad (3) \]
Now substitute Equation (3) into Equation (2):
\[ 2160 = (10b)b^2 \] \[ 2160 = 10b^3 \] \[ b^3 = \frac{2160}{10} \] \[ b^3 = 216 \] \[ b = \sqrt[3]{216} \] \[ b = 6 \]
Step 4: Finding \( a \)
Now substitute \( b \) back into Equation (3) to find \( a \):
\[ a = 10b = 10 \cdot 6 = 60 \]
Step 5: Writing the equation
Now that we have \( a \) 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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