Advanced Math / Nonlinear Functions
Exponential Functions, Growth, and Decay
Exponential functions such as $f(x)=a(b)^x$ are interpreted using the initial value $a$ and the growth or decay factor $b$. Values greater than 1 indicate growth, while values between 0 and 1 indicate decay.
Least you need to know
- In $f(x)=a(b)^x$, $a$ is the value when $x=0$.
- If $b>1$, the function grows by a factor of $b$ each time $x$ increases by 1.
- If $0<b<1$, the function decays by a factor of $b$ each time $x$ increases by 1.
- A growth factor of $1.08$ represents an 8% increase.
Key notation
- $P(t)=200(2)^t$ — An exponential growth model with initial value 200 and doubling factor 2
- $h(x)=50(0.8)^x$ — An exponential decay model with decay factor 0.8
Worked example
- If $f(x)=3(2)^x$, then $f(4)=3(2^4)=3(16)=48$.
- The coefficient 3 is multiplied by the exponential factor.
Common mistakes
- The initial value is found by substituting $x=0$, not $x=1$.
- A factor of 0.8 means a 20% decrease, not an 80% decrease.
- For $b^x$, use the given function value to solve for $b$ when the exponent is known.
How to recognize it
- The problem asks for a growth factor, decay factor, percent increase, or function value.
- The function is written in the form $a(b)^x$.
- The problem asks for a parameter using one known point on an exponential function.
Next recommended lesson
Continue through this topic with Nonlinear Function Transformations and Features.
Nonlinear Function Transformations and FeaturesRelated lessons
Keep going with nearby lessons in the same topic.