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# Exponential Functions: Exponential Growth and Decay

Jul 25, 2023

### 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

Example: Graph of f(x) = 2*

### Exponential Decay

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

Example: Graph of (1/2)*

### Applications of Exponential Growth

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 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.

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