To find the missing elements of the exponential growth equation \( Y = a(b)^x \), we can analyze the points provided:
- Points given:
- For \( x = 0, Y = 4 \)
- For \( x = 1, Y = 7.6 \)
- For \( x = 2, Y = 14.44 \)
From the point \( x = 0 \), we know that \( Y = 4 \) when \( x = 0 \). This gives us the initial value \( a \):
\[ Y = a(b)^0 \implies Y = a \implies a = 4 \]
Now we have \( Y = 4(b)^x \).
Next, let's use the next point, \( (1, 7.6) \):
\[ 7.6 = 4(b)^1 \] \[ b = \frac{7.6}{4} = 1.9 \]
Now we can write the equation as:
\[ Y = 4(1.9)^x \]
To ensure the correctness of our values, let’s check the point \( (2, 14.44) \):
\[ Y = 4(1.9)^2 = 4 \times 3.61 = 14.44 \]
This is consistent with the given point.
Thus, the complete exponential growth equation is:
\[ Y = 4(1.9)^x \]
The missing elements are:
- \( a = 4 \)
- \( b = 1.9 \)
So, in summary, the equation is \( Y = 4(1.9)^x \).
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