Question

0,4
1,7.6
2,14.44
use the graph to fill in the missing elements of the exponential growth equation
y=____(____) ^x
what is the initial value of the equation?
what is the common ratio?

Answer 1

To determine the exponential growth equation in the form \( y = a(b)^x \), we first need to identify the initial value and the common ratio from the given data points.

The points given are:
1. \( (0, 4) \)
2. \( (1, 7.6) \)
3. \( (2, 14.44) \)

### Step 1: Find the Initial Value
The initial value of the exponential equation occurs when \( x = 0 \). According to the first point \( (0, 4) \):
- The initial value \( a \) is:
\[
a = 4
\]

### Step 2: Calculate the Common Ratio
The common ratio \( b \) can be determined by analyzing the growth from one point to the next. To find \( b \), we can use the values from the first two points:

From \( (0, 4) \) to \( (1, 7.6) \):
\[
b = \frac{y(1)}{y(0)} = \frac{7.6}{4} = 1.9
\]

You can verify this by checking the growth from \( (1, 7.6) \) to \( (2, 14.44) \):
\[
b = \frac{y(2)}{y(1)} = \frac{14.44}{7.6} \approx 1.9
\]

### Final Equation
Using \( a \) and \( b \):
\[
y = 4(1.9)^x
\]

Thus, we can fill in:
- \( y = 4(1.9)^x \)
- The initial value is \( 4 \).
- The common ratio is \( 1.9 \).

### Summary
- Equation: \( y = 4(1.9)^x \)
- Initial Value: \( 4 \)
- Common Ratio: \( 1.9 \)