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.
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