Exponential Growth and Decay

Key Concepts

• Define exponential growth
• Define exponential decay
• Solve problems involving exponential growth and decay
• Find the compound interest

Exponential Growth and Decay

Exponential growth 

• The graph of the exponential function is an increasing asymptote if the value of b is greater than 1. 

Example: Graph of f(x)=2^x

• We can model exponential growth with a function f(x) = a.b^x, a>0, b>1.

Exponential growth 

• The graph of the exponential function decreases if the value of b lies between 0 and 1. 

Example: Graph of (1/2)^x

• We can model exponential decay with a function f(x) = a.b^x, a>0, 0<b<1. 

3. Applications of exponential growth 

• We can calculate the compound interest using an exponential growth function. 

Example: If Jenny invested $350 in a bank. Find the amount she will receive after 3 years if the amount was compounded quarterly at 5%? 

Solution: The principal amount is $350. 

The rate of interest is 5% or 0.05. 

The number of times per year the interest is calculated is 4. 

Compound interest = 350(1+0.05 / 4)^4×3

= 350(1+0.0125)¹²

= 350(1.0125)¹²

= 350 × 1.16075451772 

= 406.264081202 

≈ $ 406 

Exercise

• Write an exponential growth function for the initial value of 1,250, increasing at a rate of 25%.
• Write an exponential decay function for the initial value of 512, decreasing at a rate of 50%.
• What is the difference in the value after 10 years of an initial investment of $2,000 at 5% annual interest when the interest is compounded quarterly rather than annually?
• Write an exponential function to model the data in the table.

• Find the approximate value of x that makes f(x)=g(x).
• f: initial value of 200 decreasing at a rate of 7%
• g: initial value of 30 increasing at a rate of 5%

Concept Map

What have we learned

• The graph of exponential functions where, 0<b<1 is decreasing, is called Exponential Decay.
• The graph of exponential functions where, b>1 is increasing, is called Exponential Growth.