Find the growth factor b for an exponential function that passes through the points (0, 4) and (1, 20), which is _____.
We are given an exponential function in the form \(f(x) = a \cdot b^x\), where \(a\) is the initial value and \(b\) is the growth factor. We are provided with two points that the function passes through: \((0, 4)\) and \((1, 20)\). Our goal is to determine the value of the growth factor \(b\).
Using the first point \((0, 4)\), we substitute \(x=0\) and \(f(0)=4\) into the equation \(f(x) = a \cdot b^x\):
\[4 = a \cdot b^0\]
Since any non-zero number raised to the power of 0 is 1, \(b^0 = 1\). Thus, the equation simplifies to:
\[4 = a \cdot 1 = a\]
So, the initial value \(a\) is 4.
Now that we know \(a = 4\), the function is defined as \(f(x) = 4 \cdot b^x\). We use the second point \((1, 20)\), substituting \(x=1\) and \(f(1)=20\) into our updated equation:
\[20 = 4 \cdot b^1\]
Since \(b^1 = b\), we have:
\[20 = 4 \cdot b\]
To solve for the growth factor \(b\), divide both sides by 4:
\[b = \frac{20}{4} = 5\]
The growth factor is 5.
Exponential growth is a common concept used to describe how quantities increase over time, such as in population growth or compound interest. For example, if a small colony of bacteria doubles every hour, its growth is exponential. By identifying the initial amount and the growth factor from known data points, scientists can predict the size of the population at any future time. Understanding these calculations helps in fields ranging from biology and ecology to economics and finance.