Question

Use the graph to answer the question.

The graph shows an exponentially decreasing line passing through (-1, 10) and (1, 3.6)

Write the exponential equation shown in the graph.

Answer 1

To find the exponential equation that fits the points given in the problem, we can use the general form of an exponential function:

\[
y = ab^x
\]

where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base that determines the rate of decay.

Given the points (-1, 10) and (1, 3.6), we can set up a system of equations to find \( a \) and \( b \).

1. For the point (-1, 10):
\[
10 = ab^{-1} \quad \text{(Equation 1)}
\]
This simplifies to:
\[
10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b \quad \text{(Equation 1.1)}
\]

2. For the point (1, 3.6):
\[
3.6 = ab^{1} \quad \text{(Equation 2)}
\]
This can be written as:
\[
3.6 = ab \quad \text{(Equation 2.1)}
\]

Now, we can substitute Equation 1.1 into Equation 2.1:

\[
3.6 = (10b)b
\]
\[
3.6 = 10b^2
\]

Now, solve for \( b^2 \):
\[
b^2 = \frac{3.6}{10} = 0.36
\]
\[
b = \sqrt{0.36} = 0.6
\]

Now that we have found \( b \), we can use Equation 1.1 to find \( a \):
\[
a = 10b = 10 \times 0.6 = 6
\]

So, the exponential equation is:
\[
y = 6(0.6)^x
\]

Thus, the exponential equation shown in the graph is:

\[
y = 6(0.6)^x
\]