Question

Use the graph to answer the question.

The graph shows an exponentially decreasing line passing through (2, 5,000,000) and (1, 0.625)

Write the exponential equation shown in the graph.

Answer 1

To find the exponential equation represented by the points (2, 5,000,000) and (1, 0.625), we can use the general form of an exponential function, which is:

\[ y = ab^x \]

where \( a \) is the initial value (the value when \( x = 0 \)), \( b \) is the base (growth or decay factor), and \( x \) is the exponent.

1. **Use the first point (2, 5,000,000):**
\[
5,000,000 = ab^2
\]

2. **Use the second point (1, 0.625):**
\[
0.625 = ab^1
\]
Simplifying this gives:
\[
0.625 = ab
\]

Now we have two equations:

1. \( 5,000,000 = ab^2 \) (Equation 1)
2. \( 0.625 = ab \) (Equation 2)

Next, we can solve these equations simultaneously. From Equation 2, solve for \( a \):
\[
a = \frac{0.625}{b}
\]

Substitute this expression for \( a \) into Equation 1:
\[
5,000,000 = \left(\frac{0.625}{b}\right) b^2
\]
This simplifies to:
\[
5,000,000 = 0.625b
\]

Now, solve for \( b \):
\[
b = \frac{5,000,000}{0.625} = 8,000,000
\]

Now that we have \( b \), we can find \( a \). Substitute \( b \) back into Equation 2:
\[
0.625 = a(8,000,000)
\]
Thus,
\[
a = \frac{0.625}{8,000,000} = 7.8125 \times 10^{-8}
\]

Now we have the values of \( a \) and \( b \):
- \( a \approx 7.8125 \times 10^{-8} \)
- \( b = 8,000,000 \)

Finally, we can put these into the exponential function:
\[
y = (7.8125 \times 10^{-8})(8,000,000)^x
\]

This is the exponential equation represented by the graph through points (2, 5,000,000) and (1, 0.625).