To find the exponential equation for the given sequence \((0,3),(1,3.6),(2,4.32),(3,5.184)...\), we can assume the general form of the exponential equation is:
\[ y = a \cdot b^x \]
where \(a\) represents the initial value (the y-value when \(x=0\)) and \(b\) is the growth factor.
- Find the value of \(a\):
From the point \((0, 3)\): \[ y(0) = a \cdot b^0 = a = 3 \]
So we have: \[ a = 3 \]
- Find the growth factor \(b\):
Next, we can use the other points to determine \(b\). We can use the point \((1, 3.6)\): \[ y(1) = 3 \cdot b^1 = 3 \cdot b = 3.6 \] To solve for \(b\): \[ b = \frac{3.6}{3} = 1.2 \]
- Verification of \(b\):
To ensure that this value of \(b\) fits the other data points, we can calculate the next points using \(b = 1.2\).
For \(x=2\): \[ y(2) = 3 \cdot (1.2)^2 = 3 \cdot 1.44 = 4.32 \]
For \(x=3\): \[ y(3) = 3 \cdot (1.2)^3 = 3 \cdot 1.728 = 5.184 \]
Both match the given points in the sequence.
- Final equation:
Thus, the exponential equation for the sequence is:
\[ y = 3(1.2)^x \]