Definition of Exponential

Exponential models represents a group of functions that changes from one period to the next by a percent rate increase or decrease. In this text we will only example exponential functions with a constant percent rate change. 

The key concept that differentiates exponential from other functions is this periodic percent rate increase: So we define an exponential function as we do below:
 

Exponential Models are functions that change from one period to the next by a constant increase or decrease of c percent, where r is the decimal equivalence to the percent rate, c. Given the exponential model Pt, then Pt+1 = Pt(1+r): where r is the decimal representation of the percent rate and if the function is decreasing we have 1-r and 1+r for increasing functions. or , where r and k are rates and P0 is the value of P at t = 0.

All exponential models can be represented by either  or .

Consider  where r is 0.0525 or 5.25%, an equivalent model would be .

Since both functions give the same functional model we can say that .

Or 

Since  then .

So  or  (see section on logarithm), and k = 0.05117 or 5.12%

Therefore if we were to graph both functions we would get the same graphs:

 

Figure 4.1 Exponential:

Figure 4.2 Exponential:

Table 4.1 Examples of Exponential rates:
 

Exponential Models	P0	Percent Rates, %	r
10(1.06)t	10	6	0.06
1.25(2)x	1.25	100	1
¼ (3)t	¼	200	2
4 e0.045 t	4		0.04603
		So rate is 4.603%

Rates conversion to decimal 

(see section on converting from percent to decimal and from decimal to percent)

Properties of Exponential by Example

Given the exponential model:  or or 

There are a number of properties that are derived from or define the exponential model. These properties are useful when manipulating exponential functions algebraically and so must be reviewed before any serious application of the exponential model.

I. , where b, n, and m are constants:

Example 4.1a: Evaluate 

II. 

Example 4.1b: Evaluate 

III. 

Example 4.1c: Evaluate 

IV. , where a, b, and n are constants:

Example 4.1d: Evaluate 

V. 

Example 4.1e: Evaluate 

Definitions / Properties when the exponent of the exponential is .

VI.

Example 4.1f: Evaluate 

VII. 

Example 4.1g: Write as positive exponent 

VIII. , n-th root of b

Example 4.1h: Evaluate 

IX. 

Example 4.1i: Evaluate 

Exponential and Logarithm

X. If  then 

Example 4.1j: Solve for t, if 

Take logs of both sides: 

So 

Periodic and Continuous rates of Exponential Functions: 

For exponential models:  or or , the rate represented by r is often referred to as the periodic rate and the rate represented by k is often called the continuous rate. We have seen earlier that either formula may be used to represent the same function:  = = , where .

The + or - sign before the rate determine whether the models are increasing of decreasing:
 

Figure 4.3. Increasing Models: or or ,  where .

Figure 4.4 Decreasing Models: or or , where .

Table 4.2 Conversion between Periodic and Exponential Rates
 

Periodic Rate	Continuous Rate Equivalence	Conversion Formula

Example 4.2 Exponential Fine of parking ticket

If you have a problem with a periodic rate say: the monthly fine for a parking ticket after its is due is 2% of the 
balance, then the formula that represents the fine, $ after its is dues is given by the function: 
and the fine owed after 10 months on $60 or F₀ is 
 

Precalculus: Contemporary Models
by Pin D. Ling