Question

use the graph to fill in the missing elements of the exponential growth equation

graph: (0,4) (1,7.6) (2,14.44)

Answer 1

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:

1. At \( x = 0 \), \( y = 4 \)
2. At \( x = 1 \), \( y = 7.6 \)
3. 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:
\[
y = 4(1.9^x)
\]