Exponential Growth and Decay - Algebra-2

1. Fundamental Concepts

Exponential Growth: A quantity increases by a fixed percentage over each equal time period. Its growth rate accelerates (not constant in absolute terms) as time passes.
Exponential Decay: A quantity decreases by a fixed percentage over each equal time period. It gradually shrinks toward 0 but never actually reaches 0, following the fixed percentage reduction rule.
General Model: $A(t) = P(1 + r)^t$
- : The quantity after t time periods;
- P: The initial quantity (value when , called "principal" in financial contexts);
- r: Periodic growth rate (, in decimal) or decay rate (, in decimal);
- t: Number of time periods (must match the unit of r).

2. Key Concepts

Growth/Decay Factor: The term is critical. If , it is a growth factor; if , it is a decay factor (the factor is always positive).
Y-Intercept: Always , corresponding to the initial quantity.
Horizontal Asymptote: (the quantity remains positive and never reaches zero).
Domain and Range: Domain is (time cannot be negative); Range is (the quantity is always positive).

3. Examples

1. Easy

Question: A car has an initial value of 25,000 dollars and depreciates by 8% each year. Determine whether this is exponential growth or decay, and find the y-intercept of the corresponding function.
Solution:

Since the car’s value decreases by a fixed percentage (8%) annually, it is an example of exponential decay.
In the model , the initial value . The y-intercept occurs at , so substituting into the function gives . Thus, the y-intercept is .

2. Medium Difficulty (Calculate the Final Quantity)

Question: Daniel deposits 700 dollars in a savings account that earns an annual interest rate of 2.5% (following exponential growth). How much money will be in his account after 5 years (Round the result to the nearest integer)?
Solution:

Identify the variables in the model:
- Initial deposit dollars;
- Annual growth rate (converted to decimal);
- Number of time periods years (matches the annual rate unit).
Substitute the values into the exponential growth model:
Calculate step-by-step:
- First, compute the growth factor: ;
- Then calculate (using a calculator for exponentiation);
- Finally, find the final amount: .
Round to the nearest cent (standard for currency): Approximately 792.00 dollars.

3. Difficult

Question: The population of black bears in California was about 34,000 in 2014. Let represent the year 2014. By 2015, the population had increased to 35,000. Assuming the population grows exponentially, how many black bears will there be in California in 2024 (Round the result to the nearest integer) ?
Solution:

Step 1: Find the annual growth rate r
- In 2014 (), the initial population ;
- In 2015 (, 1 year after 2014), the population ;
- Substitute into the growth model: , so ;
- Solve for r: (about 2.94% annual growth rate).
Step 2: Determine the value of t for the year 2024
- Since is 2014, the number of years from 2014 to 2024 is . Thus, .
Step 3: Predict the bear population in 2024
- Substitute , , and into the model:
- Calculate the growth factor: (using a calculator);
- Compute the final population: .
Round to the nearest whole number (since population is a count of individuals): Approximately 44,754 black bears.

4. Problem-Solving Techniques

Identify Growth/Decay Type: Check if the quantity changes by a "fixed percentage" (not a fixed absolute amount). If yes, confirm growth () or decay ().
Map Variables Clearly: First, assign values to P (initial quantity), r (convert percentage to decimal), and t (ensure its unit matches r). Identify the unknown (e.g., , r, or t).
Calculation Tips: Use a calculator for exponentiation; keep 4-5 decimal places in intermediate steps to avoid rounding errors. Round final results appropriately (to the nearest cent for currency, whole number for counts).
Reverse Calculation for r or t:
- To find r: Rearrange the model to , then solve for r;
- To find t: Use logarithms: (suitable for advanced problems).