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Pre-Calculus Exponential and logarithmic functions (advanced)

Exponential models (compound, continuous, decay)

20 practice questions 0 video lessons Theory + worked examples

Exponential Functions and Models

Common Core Pre-Calculus • Standard F-IF.7e • Exponential & Logarithmic Functions

Exponential Functions and Models is the opening topic of Exponential & Logarithmic Functions in the Common Core State Standards. It is aligned to Standard F-IF.7e, which requires students to construct and interpret exponential functions and models.

An exponential function \(f(x)=a\cdot b^x\) grows when \(b>1\) and decays when \(0

Common Core Pre-Calculus › Exponential & Logarithmic Functions › Exponential Functions and Models  —  Standard F-IF.7e

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Theory

An exponential function has the variable in the exponent:

\[f(x)=a\cdot b^x,\qquad a\neq 0,\ b>0,\ b\neq 1.\]

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 \(y=0\).

Constant ratio, not constant difference. Each unit step multiplies the output by \(b\) — that is what makes growth exponential rather than linear.
Exponential growth and decay y equals 2 to the x grows, y equals one half to the x decays; both pass through 0 comma 1 and approach the x-axis. x y y=2ₓ y=(½)ₓ (0,1)
\(y=2^x\) grows and \(y=(\dfrac12)^x\) decays; both pass through \((0,1)\).
Exponential function Exponential function Exponential function f(x) = a · bˣ b > 1: growth 0 < b < 1: decay y-intercept a, asymptote y = 0
The parts of an exponential function.

Exponential function and its growth/decay factor:

\[f(x)=a\cdot b^x;\qquad \text{growth rate } r:\ b=1+r,\quad \text{decay: } b=1-r\]
f of x equals a times b to the x; growth factor is one plus r, decay is one minus r
Percent change to base: \(+3\%\) per period means \(b=1.03\); \(-15\%\) means \(b=0.85\).

How to work with exponential functions

  1. Identify \(a\) and \(b\); check \(b\) for growth vs decay.
  2. Evaluate by substituting into the exponent.
  3. Model: \(a\) is the initial amount, \(b=1\pm r\) from the percent rate.
  4. From data: \(f(0)=a\), then a second point fixes \(b\).
Example 1 — Evaluate
For \(f(x)=3\cdot 2^x\), find \(f(0)\) and \(f(3)\).
Solution

Substitute, using \(2^0=1\).

\(f(0)\)\(=\)\(3\cdot 2^0=3\)
\(f(3)\)\(=\)\(3\cdot 2^3=24\)
f of 0 is 3, f of 3 is 24
Example 2 — Growth or decay
Classify \(f(x)=5(0.85)^x\).
Solution

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.

decay of 15 percent each step
Example 3 — Population model
A town of \(8000\) grows \(3\%\) per year. Write a model and find the population after 10 years.
Solution

Growth factor \(1+0.03=1.03\).

\(P(t)\)\(=\)\(8000(1.03)^t\)
\(P(10)\)\(=\)\(8000(1.03)^{10}\approx 10{,}751\)
population after 10 years is about 10751
Example 4 — Find the base from data
An exponential function has \(f(0)=4\) and \(f(1)=12\). Find \(f(x)\).
Solution

\(f(0)=a=4\); then \(f(1)=a\cdot b=12\) gives the base.

\(4b\)\(=\)\(12\)
\(b\)\(=\)\(3\)
\(f(x)\)\(=\)\(4\cdot 3^x\)
the function is 4 times 3 to the x

Common pitfalls

The base is not the exponent. In \(a\cdot b^x\), the variable sits in the exponent, not the base.
Growth vs decay is about \(b\), not \(a\). A base above 1 grows; between 0 and 1 decays.
The asymptote is \(y=0\) (shifted if a constant is added); the graph never reaches it.

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 \(b=0.85\).

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