Question

Use the graph to fill in the missing elements of the exponential growth equation.

(1 point)
the graph has three points, (0,4) (1,7.6) (2,12.44)

Answer 1

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 \( b \).

1. For the point \( (1, 7.6) \):

\[
7.6 = 4 \cdot b^1 \implies b = \frac{7.6}{4} = 1.9
\]

2. 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).