Question

This graph shows an exponential function. graph shows two points (-1,1/20) (0,1/4)
Write an equation for the function in the form f(x)=a(b)x.
f(x)=

Answer 1

To determine the exponential function given two points \((-1, \frac{1}{20})\) and \((0, \frac{1}{4})\), we can start by using the general form of the exponential function:

\[ f(x) = a(b)^x \]

The key points we have are:

1. When \(x = 0\), \(f(0) = \frac{1}{4}\)
2. When \(x = -1\), \(f(-1) = \frac{1}{20}\)

From the point \((0, \frac{1}{4})\), we can substitute \(x = 0\) into the function:

\[
f(0) = a(b)^0 = a(1) = a
\]
Thus, we have:

\[
a = \frac{1}{4}
\]

Now we can use the second point \((-1, \frac{1}{20})\):

\[
f(-1) = a(b)^{-1} = \frac{1}{4}(b)^{-1} = \frac{1}{4 \cdot b}
\]
Setting this equal to \(\frac{1}{20}\), we get:

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

To solve for \(b\), let's cross-multiply:

\[
1 \cdot 20 = 4b \cdot 1
\]

This simplifies to:

\[
20 = 4b
\]

Now, divide by 4:

\[
b = \frac{20}{4} = 5
\]

Now that we have \(a\) and \(b\), we can substitute back:

\[
a = \frac{1}{4} \quad \text{and} \quad b = 5
\]

Thus, the exponential function is:

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

In summary, the equation of the function is:

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