Algebra 2
Exponential and logarithmic functions
Exponential functions (advanced)
20 practice questions
0 video lessons
Theory + worked examples
Exponential Functions
California Algebra 2 • Standard F-IF.7e • Exponential & Logarithmic Functions
Exponential Functions is the opening topic of Exponential & Logarithmic Functions in the California Common Core State Standards. It is aligned to Standard F-IF.7e, which requires students to graph exponential functions and interpret growth and decay in context.
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:
\[f(x)=a\cdot b^{x},\quad a\neq0,\ b>0,\ b\neq1.\]
- \(b>1\): growth. \(0<b<1\): decay.
- \(a\) is the initial value (the \(y\)-intercept).
- Horizontal asymptote \(y=0\).
The base is the multiplier applied each step.
Growth (\(b>1\)) and decay (\(0<b<1\)).
Features of \(a\cdot b^x\).
Exponential model and interest:
\[f(x)=a\cdot b^{x},\qquad A=P(1+r)^{t}\]
Growth rate \(r\) gives base \(1+r\); decay gives \(1-r\).
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.
Example 1 — Evaluate
For \(f(x)=2\cdot 3^x\), find \(f(0)\) and \(f(1)\).
Solution
Substitute each value.
| \(f(0)\) | \(=\) | \(2\cdot3^0=2\) |
| \(f(1)\) | \(=\) | \(2\cdot3^1=6\) |
Example 2 — Growth or decay
Is \(f(x)=100(0.8)^x\) growth or decay?
Solution
The base \(0.8\) is between \(0\) and \(1\).
| \(0<0.8<1\) | \(\Rightarrow\) | \(\text{decay}\) |
Example 3 — Growth factor
A quantity triples each year. Write its model from an initial \(50\).
Solution
Tripling means base \(3\).
| \(f(t)\) | \(=\) | \(50\cdot3^{t}\) |
Example 4 — Compound interest
Find the value of \(\$1000\) at \(5\%\) compounded yearly after \(3\) years.
Solution
Use \(A=P(1+r)^t\).
| \(A\) | \(=\) | \(1000(1.05)^3\) |
| \(\approx\) | \(\$1157.63\) |
Common pitfalls
\(a\cdot b^x\) is not \((ab)^x\) — only the base is raised.
Growth base \(>1\), decay base between 0 and 1.
\(b^0=1\), so \(f(0)=a\).
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 \(y=0\).
← Previous subtopic
This is the first subtopic
Next subtopic →
Logarithms (definition and evaluation)
More in Exponential and logarithmic functions