Question

This graph shows an exponential function.
Write an equation for the function in the form f(x)=a(b)x.
f(x)=

Graph point: -1, 1/20
0,1/4

Answer 1

To find the equation of the exponential function in the form \( f(x) = a(b)^x \), we can use the two points provided: \((-1, \frac{1}{20})\) and \( (0, \frac{1}{4})\).

The point \( (0, \frac{1}{4}) \) gives us a direct indication of the value of \( a \):

\[
f(0) = a(b)^0 = a = \frac{1}{4}.
\]

Now we know that \( a = \frac{1}{4} \), and we can substitute this into our function:

\[
f(x) = \frac{1}{4}(b)^x.
\]

Next, we can use the second point \((-1, \frac{1}{20})\) to find \( b \):

\[
f(-1) = \frac{1}{4}(b)^{-1} = \frac{1}{20}.
\]

This can be rewritten as:

\[
\frac{1}{4} \cdot \frac{1}{b} = \frac{1}{20}.
\]

To solve for \( b \), we can cross-multiply:

\[
20 \cdot \frac{1}{4} = 1 \cdot b.
\]

\[
\frac{20}{4} = b.
\]

\[
b = 5.
\]

Now we can substitute \( b \) back into the function:

\[
f(x) = \frac{1}{4}(5)^x.
\]

Thus, the equation for the exponential function is:

\[
\boxed{f(x) = \frac{1}{4}(5)^x}.
\]