An exponential function is a function of the form \(f(t)=ab^t\text{,}\) where \(a\) is a nonzero real number, and \(b\) is a positive real number not equal to 1. Whenever we have an expression like \(b^t\text{,}\)\(b\) is called the base and \(t\) is the exponent. The number \(e\) is often used as the base in exponential functions because it has a number of special properties (you will learn more about this in calculus). The value of \(e\) is about 2.718.
Note: In an exponential function, the variable is in the exponent. For example \(f(x)=2^x\) is an exponential function, but \(g(x)=x^2\) is not an exponential function because the exponent is not a variable.
Company 1: For each day the laptop is late, you owe an additional $5. On day 1, your total late penalty is $5. On day 2, your total late penalty is $10. On day 3, your total late penalty is $15, and so on.
Company 2: For each day the laptop is late, your penalty doubles from the previous day. On day 1, your late penalty starts at $0.25. On day 2, your late penalty doubles to $0.50. On day 3, the late penalty doubles to $1, and so on.
In Desmos, graph \(f\) and create sliders for \(a\text{,}\) and \(b\text{.}\) What role do each of these constants play? Include sketches and verbal descriptions to help explain the role of each constant.
\(f(x)=2^x\) is an exponential function. A table of values for \(f(x)\) is shown in Table 2.1.3. Recall that a negative exponent means taking the reciprocal of the positive exponent, so \(f(-1)=2^{-1}=\frac{1}{2^1}=\frac{1}{2}\text{.}\) Graph \(f\) on the domain \(-1 \leq x \leq 5\text{.}\) Describe the shape of the graph.
Let \(g(x)=10\cdot 2^x\text{,}\)\(h(x)=20^x\text{,}\)\(k(x)=10^x 2^x\text{,}\) and \(m(x)=(10\cdot 2)^x\text{.}\) By graphing these functions, decide which, if any, of these are really the same function.
Consider \(2^x 3^y\text{,}\)\(6^{xy}\text{,}\) and \(6^{x+y}\text{.}\) By plugging in pairs of values for \(x\) and \(y\text{,}\) decide whether any of these are the same.
