BackC2-C3
Study Guide - Smart Notes
Tailored notes based on your materials, expanded with key definitions, examples, and context.
Exponential Functions
Definition and Identification
An exponential function is a function of the form:
where a and b are constants, a ≠ 0, b > 0, and b ≠ 1. The variable x appears in the exponent.
Exponential growth occurs when .
Exponential decay occurs when .
To determine if a function is exponential, check if the variable is in the exponent and the base is a positive constant not equal to 1.
Examples: Identifying Exponential Functions
→ Exponential (base 2)
→ Not exponential (variable is the base)
→ Exponential (base 2, a = 3)
→ Exponential (can be rewritten as )
→ Exponential (decay, base )
→ Not exponential (exponent is not linear in x)
Properties of Exponential Functions
Domain:
Range:
y-intercept:
Horizontal asymptote:
Exponential growth: function increases as increases.
Exponential decay: function decreases as increases.
Comparing Linear and Exponential Functions in Tables
In a linear function, constant additions in produce constant additions in .
In an exponential function, constant additions in produce constant multiplications in .
x | Linear f(x) | Exponential f(x) |
|---|---|---|
0 | 2 | 3 |
1 | 5 | 6 |
2 | 8 | 12 |
3 | 11 | 24 |
Additional info: In the exponential column, each increase in x multiplies y by 2.
Graphing Exponential Functions
Basic Shapes
Exponential Growth: with (curve rises rapidly as increases)
Exponential Decay: with (curve falls rapidly as increases)
All exponential functions have a horizontal asymptote at unless shifted vertically.
Example: Table and Graph
x | f(x) = 3 \cdot 2^x |
|---|---|
-2 | 0.75 |
-1 | 1.5 |
0 | 3 |
1 | 6 |
2 | 12 |
Plotting these points shows the rapid increase characteristic of exponential growth.
Modeling with Exponential Functions
Population Growth and Decay
General formula for exponential growth/decay:
= initial value
= growth (or decay) rate (as a decimal)
= time
Example: A town has a population of 1000 people. If the population increases by 5% per year:
If the population decreases by 12% per year:
Finding an Exponential Equation Through Two Points
Procedure
Given two points and , assume .
Substitute both points to get two equations:
Divide the equations to solve for :
Solve for , then substitute back to find .
Example: Through points (2, 12) and (4, 27):
So, .
Transformations of Exponential Functions
General Form and Effects
The general form is:
h: horizontal shift (right if , left if )
k: vertical shift (up if , down if )
a: vertical stretch/compression and reflection (if )
Examples of Transformations
: Shift right 1 unit
: Shift up 4 units
: Reflect over x-axis
: Shift left 3 units
Key Features to Identify
y-intercept: Set
Horizontal asymptote:
Domain:
Range: if ; if
Practice Problems
Determine which functions are exponential.
Model population growth or decay using exponential functions.
Find the equation for an exponential function passing through two points.
Describe transformations of exponential functions and sketch their graphs.
Additional info: These notes cover the identification, properties, graphing, modeling, and transformation of exponential functions, which are essential topics in College Algebra.
