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Exponential Growth and Decay

Sep 14, 2022

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)=2x

• We can model exponential growth with a function f(x) = a.bx, 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.bx, 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%

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.

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