Exponential models (compound, continuous, decay)
Exponential Functions and Models
Exponential Functions and Models is the opening topic of Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(N), P.5(I), which requires students to analyze situations modeled by exponential functions and solve exponential equations.
An exponential function \(f(x)=a\cdot b^x\) grows when \(b>1\) and decays when \(0
Theory
An exponential function has the variable in the exponent:
The base \(b\) sets the behavior: \(b>1\) gives growth, \(0<b<1\) gives decay. Every such graph passes through \((0,a)\), has domain all reals, range \(y>0\) (when \(a>0\)), and a horizontal asymptote \(y=0\).
Exponential function and its growth/decay factor:
How to work with exponential functions
- Identify \(a\) and \(b\); check \(b\) for growth vs decay.
- Evaluate by substituting into the exponent.
- Model: \(a\) is the initial amount, \(b=1\pm r\) from the percent rate.
- From data: \(f(0)=a\), then a second point fixes \(b\).
Substitute, using \(2^0=1\).
| \(f(0)\) | \(=\) | \(3\cdot 2^0=3\) |
| \(f(3)\) | \(=\) | \(3\cdot 2^3=24\) |
The base \(0.85\) is between 0 and 1, so the function decays.
| \(0<0.85<1\) | \(\Rightarrow\) | \(\text{decay}\) |
It decreases by \(15\%\) each step.
Growth factor \(1+0.03=1.03\).
| \(P(t)\) | \(=\) | \(8000(1.03)^t\) |
| \(P(10)\) | \(=\) | \(8000(1.03)^{10}\approx 10{,}751\) |
\(f(0)=a=4\); then \(f(1)=a\cdot b=12\) gives the base.
| \(4b\) | \(=\) | \(12\) |
| \(b\) | \(=\) | \(3\) |
| \(f(x)\) | \(=\) | \(4\cdot 3^x\) |
Common pitfalls
Frequently asked questions
What is an exponential function?
A function \(f(x)=a\cdot b^x\) with the variable in the exponent. It grows if \(b>1\) and decays if \(0<b<1\).
How do you tell growth from decay?
Look at the base: greater than 1 is growth, between 0 and 1 is decay.
What is the horizontal asymptote of an exponential function?
\(y=0\) for \(f(x)=a\cdot b^x\); adding a constant \(k\) shifts it to \(y=k\).
How do you find the base from a percent rate?
Add or subtract the rate from 1: \(+3\%\) gives \(b=1.03\), \(-15\%\) gives \(b=0.85\).