Question

The graph of an exponential function passes through the points (–1,10) and (1,2/5)
. Write an equation for the function in the form y=a(b)^x.
y=

Answer 1

To find an exponential function in the form \( y = a(b)^x \) that passes through the points (-1, 10) and (1, 2/5), we first use the two points to create a system of equations.

1. For the point (-1, 10):
\[
10 = a(b)^{-1} \quad \Rightarrow \quad 10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b
\]

2. For the point (1, 2/5):
\[
\frac{2}{5} = a(b)^{1} \quad \Rightarrow \quad \frac{2}{5} = ab \quad \Rightarrow \quad a = \frac{2}{5b}
\]

Now we have two expressions for \( a \):
1. \( a = 10b \)
2. \( a = \frac{2}{5b} \)

Set these equal to each other:
\[
10b = \frac{2}{5b}
\]

To solve for \( b \), first multiply both sides by \( 5b \) to eliminate the fraction:
\[
50b^2 = 2
\]
\[
b^2 = \frac{2}{50} = \frac{1}{25} \quad \Rightarrow \quad b = \frac{1}{5} \quad (\text{since } b > 0)
\]

Now, substitute \( b \) back into the equation for \( a \):
\[
a = 10b = 10 \cdot \frac{1}{5} = 2
\]

Thus, we have \( a = 2 \) and \( b = \frac{1}{5} \).

Now we can write the equation of the exponential function:
\[
y = 2\left(\frac{1}{5}\right)^x
\]

This can also be rewritten as:
\[
y = 2 \cdot 5^{-x}
\]

Therefore, the equation of the exponential function is:
\[
\boxed{y = 2\left(\frac{1}{5}\right)^x}
\]