Question

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

0,4. 1,7.6. 2,14.44

Y = _(_)^x

Answer 1

To find the missing elements of the exponential growth equation \( Y = a(b)^x \), we can analyze the points provided:

1. **Points given**:
- For \( x = 0, Y = 4 \)
- For \( x = 1, Y = 7.6 \)
- For \( x = 2, Y = 14.44 \)

From the point \( x = 0 \), we know that \( Y = 4 \) when \( x = 0 \). This gives us the initial value \( a \):

\[
Y = a(b)^0 \implies Y = a \implies a = 4
\]

Now we have \( Y = 4(b)^x \).

Next, let's use the next point, \( (1, 7.6) \):

\[
7.6 = 4(b)^1
\]
\[
b = \frac{7.6}{4} = 1.9
\]

Now we can write the equation as:

\[
Y = 4(1.9)^x
\]

To ensure the correctness of our values, let’s check the point \( (2, 14.44) \):

\[
Y = 4(1.9)^2 = 4 \times 3.61 = 14.44
\]

This is consistent with the given point.

Thus, the complete exponential growth equation is:

\[
Y = 4(1.9)^x
\]

The missing elements are:
- \( a = 4 \)
- \( b = 1.9 \)

So, in summary, the equation is \( Y = 4(1.9)^x \).