Exponential Functions: Exponential Growth and Decay

Exponents

Repeated multiplication can be represented in more than one way.

You can use an exponent to write the repeated multiplication of a number.

Rational Exponents

The expressions with exponents that are rational numbers are called rational exponents (also called fractional exponents).

Exponential Functions

The product of an initial amount and a constant ratio raised to a power is an exponential function.

Exponential functions are modeled using f(x) = a.b*, where a is a non-zero constant, b>0, and b not equal to 0.

Exponential Growth

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

Example: Graph of f(x) = 2*

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

Exponential Decay

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

Example: Graph of (1/2)*

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

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 We Have 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.