1. COMPARING EXPONENTIAL AND LINEAR FUNCTIONS
4.2
COMPARING EXPONENTIAL AND LINEAR FUNCTIONS

2. Identifying Linear and Exponential Functions From a Table
Example
Notice that the value of x changes by equal steps of x = 5. The function f could be linear because the difference between consecutive values of f(x) is constant: f(x) increases by 15 each time x increases by 5.
On the other hand, the difference between consecutive values of g(x) is not constant. However, the ratio of consecutive values of g(x) is constant: 1200/1000 = 1.2, 1440/1200 = 1.2, etc. Thus, each time x increases by 5, the value of g(x) increases by a factor of 1.2. This pattern of constant ratios is indicative of exponential functions.
Two functions, one linear and one exponential x 20 25 30 35 40 45
f(x) 60 75 90
105
g(x)
1000
1200
1440
1728
2073.6

3. Identifying Linear and Exponential Functions From a Table
For a table of data that gives y as a function of x and in which x is constant: • If the difference of consecutive y-values is constant, the table could represent a linear function. • If the ratio of consecutive y-values is constant, the table could represent an exponential function.

4. Finding a Formula for an Exponential Function
Example
continued
If f(x) is a linear function, knowing that f(x) increases by 15 each time x increases by 5 tells us that the slope of the line is 3. Then the point-slope equation gives y – 30 = 3 (x – 20) or f(x) = 3 x – 30.
Two functions, one linear and one exponential x 20 25 30 35 40 45
f(x) 60 75 90
105
g(x)
1000
1200
1440
1728
2073.6

9. Linear: f(x) increases by a constant of 10 ; y = 10x + 65
(b) Exponential: g(x) increases by a percentage of 50% ; y = 400(1.5)t
Solution:

11. Exponential Growth Will Always Outpace Linear Growth in the Long Run
Consider the linear function f(x) = 1000x versus the exponential function g(x) = 1.1x
g(x) = 1.1x
f(x) = 1000x