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Haberman / Kling MTH 111c
Section II: Exponential and Logarithmic Functions
Module 3: Comparing Linear and Exponential Functions
EXAMPLE 1: The table below gives values for the functions f and. Determine which is a linear function and which is an exponential function.
g
x โ5 0 5 10 15 f ( x ) (^) 7 22 37 52 67 g x ( )^116 2 64 2048
Before deciding, letโs study tables representing a functions that we know are either linear or exponential.
EXAMPLE 1a: Let k x ( ) = 4 x + 3 (a linear function)
x โ2 โ1 0 1 2 3 8 k x ( ) (^) โ5 โ1 3 7 11 15 35
Notice that each time we make the same change in the x -value, the corresponding y - values have a constant difference (i.e., when you subtract the numbers, the result is always the same). For example, if we increase x -values by 1 , the difference of corresponding y -values is always 4 :
Change x from โ1 to 0 : ( ) 0 ( ) 3 ( 4
k โ k 1 = โ โ1)
Change x from 0 to 1 : ( ) 1 ( ) 0 7 3 4
k โ k = โ
Change x from 1 to 2 : ( ) 2 ( ) 1 1 1 7 4
k โ k = โ
Note that we see this same behavior (where corresponding y -values have a constant difference ) with any constant change in x -values. Letโs see what happens if we change our x -values by 5 :
Change x from โ2 to 3 : ( ) 3 ( 2 ) 1 ( 5) 20
k โ k = 5 โ โ
Change x from 3 to 8 : ( ) 8 ( ) 3 3 2
k โ k = 5 โ 1
So if we change our x -values by 5 the difference of corresponding y -values is 20.
In general, if a function is linear and you make a constant change in x -values, the difference of corresponding y -values is constant.
EXAMPLE 1b: Let j x ( ) = 2 3โ
x (an exponential function)
x โ2 โ1 0 1 2 3 8 j x ( )^2 9 2 3 2 6 18 54
Notice that each time we make the same change in the x -value, the corresponding y - values have a constant ratio (i.e., when you divide the y -values, the result is always the same). For example, if we increase x -values by 1 , the ratio of corresponding y -values is 3 :
Change x from โ1 to 0 :
(^23)
j j = =
Change x from 0 to 1 : 1 0
j j = = Change x from 1 to 2 : ( ) (^18) ( )
j j
Change x from 10 to 15 : ( 15 ) ( 10 ) 67 5 5
f โ f = โ
Since the difference of the corresponding y -values is constant, f is a linear function.
Now weโll study. When we change our x -values by 5 the ratio of corresponding y - values is 32 :
y = g x ( )
Change x from โ5 to 0 :
(^116)
g g =
Change x from 0 to 5 : ( ) (^64) ( )
g g = =
Change x from 5 to 10 : ( ) (^2048) ( )
g g
Change x from 10 to 15 : ( 15 ) (^65536) ( ) 32
g g = =
Since the ratio of the corresponding y -values is constant, g is an exponential function.
Notice that it isnโt correct to say, โExponential functions increase faster than linear functions.โ In fact, there is an interval during which the linear function increases faster than the exponential function!
EXAMPLE 2: Find algebraic rules for both f and g from EXAMPLE 1.
First, letโs find the rule for f. Since f is linear, we know that f ( ) x = mx + b. Since (0, 22) satisfies f , we know that f ( ) x = mx + 22. We can use the points and to find the slope, m :
m = โ โ =
Thus, f ( ) x = 3 x + 22.
Now weโll find the rule for g. Since g is exponential, we know that^ g x ( )^^ =^ a b โ
^ x. Since
(0, 2) satisfies g , we know that g x ( ) = 2 โ
bx. We can use any other ordered pair that satisfies g to find b :
5 5
g b b b
Thus, g x ( ) = 2 2โ
x.
