Exponential functions (advanced)
Exponential Functions
Exponential Functions is the opening topic of Exponential & Logarithmic Functions in the Texas Essential Knowledge and Skills (Algebra II, §111.40). It is aligned to Standard 2A.5(A), which requires students to determine the effects of parameter changes on the graph of an exponential function.
An exponential function \(f(x)=a\cdot b^x\) grows when \(b>1\) and decays when \(0<b<1\), with \(y=0\) as its asymptote.
Theory
An exponential function has the variable in the exponent:
- \(b>1\): growth. \(0<b<1\): decay.
- \(a\) is the initial value (the \(y\)-intercept).
- Horizontal asymptote \(y=0\).
Exponential model and interest:
How to use an exponential model
- Identify the initial value \(a\).
- Find the base \(b\) (growth or decay factor).
- Write \(f(x)=a\cdot b^x\).
- Substitute to evaluate or predict.
Substitute each value.
| \(f(0)\) | \(=\) | \(2\cdot3^0=2\) |
| \(f(1)\) | \(=\) | \(2\cdot3^1=6\) |
The base \(0.8\) is between \(0\) and \(1\).
| \(0<0.8<1\) | \(\Rightarrow\) | \(\text{decay}\) |
Tripling means base \(3\).
| \(f(t)\) | \(=\) | \(50\cdot3^{t}\) |
Use \(A=P(1+r)^t\).
| \(A\) | \(=\) | \(1000(1.05)^3\) |
| \(\approx\) | \(\$1157.63\) |
Common pitfalls
Frequently asked questions
What makes a function exponential?
The variable appears in the exponent, as in \(a\cdot b^x\).
When is it growth versus decay?
Growth if \(b>1\); decay if \(0<b<1\).
What does \(a\) represent?
The initial value, the \(y\)-intercept.
What is the asymptote of an exponential function?
The horizontal line \(y=0\).