Exponential Functions

Exponential functions model situations in which a quantity grows or decays at a rate proportional to its current value. Unlike linear functions (constant rate of change) or polynomial functions (variable but finite rates), exponential functions exhibit multiplicative change over equal intervals.

They are fundamental in modeling:

• population growth

• compound interest

• radioactive decay

• spread of disease

• depreciation

Definition of an Exponential Function

An exponential function has the general form

where:

• a≠0 is the initial value (y-intercept)

• b>0, b≠1 is the base

• x is the exponent

Growth vs. Decay

Exponential Growth

If b>1

the function represents exponential growth.

• Values increase as $x$ increases

• The graph rises from left to right

Example:

This represents a 20% increase per unit.

Exponential Decay

If 0<b<1

the function represents exponential decay.

• Values decrease as $x$ increases

• The graph falls from left to right

Example:

This represents a 15% decrease per unit.

Key Characteristics of Exponential Functions

Domain and Range

• Domain: (−∞,∞)

• Range:

  • (0,∞) if a>0

  • (−∞,0) if a<0

Intercepts

• y-intercept: (0,a)

• x-intercept: None (unless the function is shifted)

Horizontal Asymptote

The x-axis (y=0) is a horizontal asymptote.

• The graph approaches but never reaches y=0

• Vertical shifts move the asymptote accordingly

Rate of Change in Exponential Functions

Unlike linear functions, exponential functions have a constant percent rate of change, not a constant difference.

For a base b:

Percent change = (b – 1) x 100%

Examples:

• b=1.08⇒8% growth

• b=0.92⇒8% decay

Transformations of Exponential Functions

The general transformed form is:

$$f(x)=a\cdot b^{(x-h)}+k$$

Effects:

• a: vertical stretch / reflection

• $h$: horizontal shift

• $k$: vertical shift (changes asymptote to y=k)

Evaluating and Comparing Exponential Functions

Comparing Growth Rates

• Larger base $b$ → faster growth

• Compare functions by:

  • base value

  • percent rate

  • long-term behavior

Long-Term Behavior

As x→∞:

• Growth functions → ∞

• Decay functions → asymptote

As x→−∞:

• Growth functions → asymptote

• Decay functions → ∞

Real-World Modeling

Exponential Model

$$f(x)=a(1+r)^t$$

where:

• a: initial amount

• r: growth or decay rate

• t: time

Compound Interest

$$A=P(1+\frac{r}{n})^nt$$

This is a specific application of exponential growth.

Common Errors to Avoid

• Confusing exponential growth with linear growth

• Interpreting base incorrectly

• Assuming exponential functions have x-intercepts

• Ignoring asymptotic behavior

• Misreading percent increase vs. base value

AP Exam Focus

On the AP Precalculus exam, students should be able to:

• Identify exponential growth and decay

• Interpret parameters in context

• Compare exponential models

• Analyze transformations

• Explain long-term behavior verbally and algebraically

Summary

Exponential functions:

• model multiplicative change

• have constant percent rates

• exhibit asymptotic behavior

• are essential for real-world modeling

Understanding exponential functions is critical for success in AP Precalculus and forms a foundation for calculus concepts such as limits and derivatives.