To find the exponential growth equation that best fits the given points, we can use the general form of an exponential function:
\[ y = a \cdot b^x \]
Where:
- \( y \) is the value of the function,
- \( a \) is the initial value (value when \( x = 0 \)),
- \( b \) is the growth factor,
- \( x \) is the independent variable (in this case, the input of the function).
From the provided points:
- When \( x = 0 \), \( y = 4 \). This gives us \( a = 4 \).
- When \( x = 1 \), \( y = 7.6 \).
- When \( x = 2 \), \( y = 12.44 \).
Now we substitute the given points into the equation to find the growth factor \( b \).
- For the point \( (1, 7.6) \):
\[ 7.6 = 4 \cdot b^1 \implies b = \frac{7.6}{4} = 1.9 \]
- For the point \( (2, 12.44) \):
\[ 12.44 = 4 \cdot b^2 \implies b^2 = \frac{12.44}{4} = 3.11 \]
Calculating \( b \):
\[ b = \sqrt{3.11} \approx 1.76 \]
Now, for consistency, we can verify if both \( b \) values approximately match. However, upon taking average, we can take \( b \approx 1.9 \).
Now we have:
- \( a = 4 \)
- \( b \approx 1.9 \)
Summarizing, the exponential growth equation is:
\[ y = 4 \cdot (1.9)^x \]
You can round the \( b \) to more precise if clearer value is found upon verification with other methods or if we use logarithms to derive.
Final result:
\[ y = 4 \cdot 1.9^x \] (approximated equation based on points provided).