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:
- When \(x = 0\), \(f(0) = \frac{1}{4}\)
- 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:
\[ f(x) = \frac{1}{4}(5^x) \]