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Comparing Linear & Exponential Functions: Identifying from Tables, Study notes of Mathematics

An in-depth analysis of the differences between linear and exponential functions using examples. It explains how to identify linear and exponential functions based on the constant difference or ratio of corresponding y-values when x-values change. Two examples are given, and the algebraic rules for each function are derived.

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

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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 โ€“5 0 5 10 15
()
f
x 7 22 37 52 67
()gx 16
1 2 64 2048 65536
Before deciding, letโ€™s study tables representing a functions that we know are either linear or
exponential.
EXAMPLE 1a: Let (a linear function)
() 4 3kx x=+
x โ€“2 โ€“1 0 1 2 3 8
()kx โ€“5 โ€“1 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 โ€“1 to 0:
() ( ) 3 (0
4
11)kk
โˆ’
=โˆ’โˆ’
=
โˆ’
Change x from 0 to 1:
10() ( ) 7 3
4
kk
โˆ’
=โˆ’
=
Change x from 1 to 2:
() () 1 7211
4
kk
โˆ’
=โˆ’
=
pf3
pf4
pf5

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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 โ€“5 0 5 10 15 f ( x ) (^) 7 22 37 52 67 g x ( )^116 2 64 2048

Before deciding, letโ€™s 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 โ€“2 โ€“1 0 1 2 3 8 k x ( ) (^) โ€“5 โ€“1 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 โ€“1 to 0 : ( ) 0 ( ) 3 ( 4

k โˆ’ k 1 = โˆ’ โˆ’1)

Change x from 0 to 1 : ( ) 1 ( ) 0 7 3 4

k โˆ’ k = โˆ’

Change x from 1 to 2 : ( ) 2 ( ) 1 1 1 7 4

k โˆ’ k = โˆ’

Note that we see this same behavior (where corresponding y -values have a constant difference ) with any constant change in x -values. Letโ€™s see what happens if we change our x -values by 5 :

Change x from โ€“2 to 3 : ( ) 3 ( 2 ) 1 ( 5) 20

k โˆ’ k = 5 โˆ’ โˆ’

Change x from 3 to 8 : ( ) 8 ( ) 3 3 2

k โˆ’ k = 5 โˆ’ 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 โ€“2 โ€“1 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 โ€“1 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 โˆ’ f = โˆ’

Since the difference of the corresponding y -values is constant, f is a linear function.

Now weโ€™ll study. When we change our x -values by 5 the ratio of corresponding y - values is 32 :

y = g x ( )

Change x from โ€“5 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 isnโ€™t correct to say, โ€œExponential 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, letโ€™s 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 = โˆ’ โˆ’ =

Thus, f ( ) x = 3 x + 22.

Now weโ€™ll 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.