To find the exponential growth equation of the form \( y = a \cdot b^x \), we can start by using the points provided.
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Identify Points:
- Starting point (0, 4): \( y = 4 \) when \( x = 0 \)
- Second point (1, 7.6): \( y = 7.6 \) when \( x = 1 \)
- Third point (2, 14.44): \( y = 14.44 \) when \( x = 2 \)
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Using the Starting Point: From the start point (0, 4): \[ y = a \cdot b^0 = a \cdot 1 = a \] So, \( a = 4 \).
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Finding \( b \): Now we can use the point (1, 7.6): \[ 7.6 = 4 \cdot b^1 \] \[ b = \frac{7.6}{4} = 1.9 \]
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Verify with the Third Point: Now we can check to see if this value of \( b \) holds with the point (2, 14.44): \[ y = 4 \cdot (1.9)^2 \] Calculate \( (1.9)^2 \): \[ (1.9)^2 = 3.61 \] Therefore: \[ y = 4 \cdot 3.61 = 14.44 \] This confirms our calculations.
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Final Exponential Growth Equation: Putting it all together, the exponential growth equation is: \[ y = 4 \cdot (1.9)^x \]
So, the complete equation is: \[ y = 4(1.9)^x \]
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