Question

Use the graph to fill in the missing elements of the exponential growth equation. (1 point) y=y=()x

Graph: starting point-(0,4)
Second point- (1,7.6)
Third point-(2,14.44)

Answer 1

To find the exponential growth equation of the form \( y = a \cdot b^x \), we can start by using the points provided.

1. **Identify Points:**
- Starting point (0, 4): \( y = 4 \) when \( x = 0 \)
- Second point (1, 7.6): \( y = 7.6 \) when \( x = 1 \)
- Third point (2, 14.44): \( y = 14.44 \) when \( x = 2 \)

2. **Using the Starting Point:**
From the start point (0, 4):
\[
y = a \cdot b^0 = a \cdot 1 = a
\]
So, \( a = 4 \).

3. **Finding \( b \):**
Now we can use the point (1, 7.6):
\[
7.6 = 4 \cdot b^1
\]
\[
b = \frac{7.6}{4} = 1.9
\]

4. **Verify with the Third Point:**
Now we can check to see if this value of \( b \) holds with the point (2, 14.44):
\[
y = 4 \cdot (1.9)^2
\]
Calculate \( (1.9)^2 \):
\[
(1.9)^2 = 3.61
\]
Therefore:
\[
y = 4 \cdot 3.61 = 14.44
\]
This confirms our calculations.

5. **Final Exponential Growth Equation:**
Putting it all together, the exponential growth equation is:
\[
y = 4 \cdot (1.9)^x
\]

So, the complete equation is:
\[
y = 4(1.9)^x
\]