To derive the exponential growth equation from the given points, we will start with the general form of an exponential growth equation, which is:
\[ y = ab^x \]
where:
- \( y \) is the value at time \( x \),
- \( a \) is the initial value (the value when \( x = 0 \)),
- \( b \) is the growth factor,
- \( x \) is the time.
From the graph, we have the following points:
- At \( x = 0 \), \( y = 4 \)
- At \( x = 1 \), \( y = 7.6 \)
- At \( x = 2 \), \( y = 14.44 \)
Step 1: Determine \( a \)
From the first point \( (0, 4) \):
- When \( x = 0 \), \( y = 4 \), so \( a = 4 \).
Step 2: Determine \( b \)
We can use the second point \( (1, 7.6) \): \[ 7.6 = 4b^1 \] Dividing both sides by 4: \[ b = \frac{7.6}{4} = 1.9 \]
Now we can confirm this by checking the third point \( (2, 14.44) \): \[ y = 4b^2 \] Substituting \( b = 1.9 \): \[ y = 4(1.9^2) \] Calculating \( 1.9^2 \): \[ 1.9^2 = 3.61 \] Now calculate \( y \): \[ y = 4 \times 3.61 = 14.44 \]
Final Equation
Thus, the exponential growth equation is:
\[ y = 4(1.9^x) \]
In summary:
- Initial value \( a = 4 \)
- Growth factor \( b = 1.9 \)
The final equation is: \[ y = 4(1.9^x) \]