Exponential and Linear Functions

Identifying Function Types from Data

When analyzing a table of input-output values, it is important to determine whether the relationship is linear, exponential, or neither. This can be done by examining consecutive differences and ratios.

• Linear Function: The difference between consecutive outputs (y-values) is constant when the input (x-values) increases by a fixed amount.

• Exponential Function: The ratio between consecutive outputs is constant when the input increases by a fixed amount.

• Neither: If neither the differences nor the ratios are constant, the function is neither linear nor exponential.

Example Table:

Since neither the differences nor the ratios are constant, the function is neither linear nor exponential.

Exponential Growth and Decay

Exponential functions model situations where a quantity grows or decays by a constant percentage rate per unit time. The general form is:

• Growth: where

• Decay: where

Key Terms:

• Initial Value (a): The starting amount at .

• Base (b): The growth (if ) or decay (if $0

Example: A population of 10 million increases by 5% per year:

• Formula:

• After 1 year: million

• After 2 years: million

Doubling Time: The time it takes for a quantity to double in size is given by:

• For , years

Exponential Decay in Applications

Exponential decay models are used in contexts such as radioactive decay, cooling, and chemical concentration reduction.

• Example: Chlorine concentration in a pool decreases by 15% per day.

• Formula: where is in ppm and is days after treatment.

• After 5 days: ppm

To find when the concentration drops below a certain threshold (e.g., 2 ppm), solve for .

Converting Between Exponential Forms

Exponential functions can be written in different forms:

• Standard:

• Continuous:

To convert to :

• Set , so

• Example: becomes

Solving Exponential Equations

To solve equations involving exponentials, use properties of exponents and logarithms.

• Example:

• Rewrite: , so

• For , rearrange and solve for using logarithms or algebraic manipulation.

Modeling with Exponential Functions

Given two points, you can determine the formula for an exponential function .

• Given and , solve for and .

• Example: If when and when :

• Set up equations: ,

• Divide:

• Substitute back:

• So,

Atmospheric Pressure and Exponential Decay

Atmospheric pressure decreases exponentially with altitude. The general model is:

• where is pressure, is altitude, is pressure at sea level, and is the decay factor per km.

Example Table:

• Find :

• Formula:

• At 8.85 km (Mt. Everest): mbar

Comparing Linear and Exponential Growth

Linear growth adds a constant amount each period, while exponential growth multiplies by a constant factor.

• Linear:

• Exponential:

Example: Country A grows linearly, Country B grows exponentially:

• Formulas: ,

• Exponential growth will eventually surpass linear growth.

Interpreting Exponential Models

Given several exponential models, you can compare growth/decay rates and initial values.

• Growth:

• Decay:

• Initial Value: The coefficient

Example:

• Largest initial value: (town iii)

• Smallest initial value: (town ii)

• Fastest growth: (town ii, 10%)

• Fastest decay: (town iii, 5%)

Summary Table: Linear vs. Exponential Functions

Key Takeaways

• Check for constant differences (linear) or ratios (exponential) in data tables.

• Use for exponential models; for growth, for decay.

• Convert between forms using for .

• Exponential growth will eventually surpass linear growth given enough time.

• Doubling time and half-life are important concepts in exponential models.