To find the exponential equation of the given sequence (0, 4), (1, 7.6), (2, 14.44), we'll use the general form of an exponential function, which is:
\[ y = a \cdot b^x \]
where \( a \) is the initial value when \( x = 0 \), and \( b \) is the growth factor.
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Find \( a \): From the first point (0, 4), we can see that when \( x = 0 \):
\[ y = a \cdot b^0 = a \]
Thus, \( a = 4 \).
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Use another point to find \( b \): Now we'll use the point (1, 7.6) to find \( b \):
\[ 7.6 = 4 \cdot b^1 \] \[ b = \frac{7.6}{4} = 1.9 \]
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Check with the next point (2, 14.44): Now we will check whether this value of \( b \) works with the third point (2, 14.44):
\[ y = 4 \cdot (1.9)^2 \] \[ y = 4 \cdot 3.61 = 14.44 \]
This matches the third point perfectly.
Thus, the exponential equation for the sequence is:
\[ y = 4 \cdot (1.9)^x \]