comparing linear and exponential behavior
Module 4 : Investigation 2
MAT 170 | Precalculus
September 23, 2016
question 1
iTech Device : Current sale value is $300. Value will decrease 30%
per year.
Dynasystems Device : Current sale value is $300. Value will decrease $45 per year.
(a) Find the resale value (in dollars) for both devices 1 year from
now. How did you determine this value ?
(b) Without performing any calculations, predict which device will
have the greater resale value 6 years from now ?
2
question 1
iTech Device : Current sale value is $300. Value will decrease 30%
per year.
Dynasystems Device : Current sale value is $300. Value will decrease $45 per year.
1
1
1
1
1
1
210
147
102.9
72.03
50.421
35.2947
-90
1
-63
-44.10
-30.87
1
1
1
-21.609
-15.1263
1
1
255
210
165
120
75
30
-45
-45
-45
-45
-45
-45
3
question 2
iTech Device : Current sale value is $300. Value will decrease 30%
per year.
Dynasystems Device : Current sale value is $300. Value will decrease $45 per year.
(a)(i) Define a function f that determines the value of the iTech
device in terms of the number of years from now, t.
(a)(ii) Define a function g that determines the value of the
Dynasystems device in terms of the number of years from now, t.
(b) Explain what each part of the expression giving the function rule
represents in the given context.
4
question 4
ǓèƐŴNJ 1ƐʟǓŴƐ ٔ )ʂɏɏƐȌɬ ɟňǵƐ ʟňǵʂƐ Ǔɟ ‫׫׫׮ڥ‬ٛ ēňǵʂƐ ʣǓǵǵ žƐŴɏƐňɟƐ ‫׫׮‬%
ɆƐɏ ʬƐňɏٛ
1ʬȌňɟʬɟɬƐȇɟ 1ƐʟǓŴƐ ٔ )ʂɏɏƐȌɬ ɟňǵƐ ʟňǵʂƐ Ǔɟ ‫׫׫׮ڥ‬ٛ ēňǵʂƐ ʣǓǵǵ žƐ‫ځ‬
The table on the left has
a column
giving the value of s = f(t) and
ŴɏƐňɟƐ
$‫ װׯ‬ɆƐɏ ʬƐňɏٛ
well as ∆s.
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׫׬׭‬
‫ײׯ׬‬
‫׭׫׬‬ٛ‫״‬
‫׭ײ‬ٛ‫׮׫‬
‫׫װ‬ٛ‫׬׭ׯ‬
‫װ׮‬ٛ‫ײׯ״׭‬
‫׫״ځ‬
‫׬‬
‫׮ױځ‬
‫ׯׯځ‬ٛ‫׫׬‬
‫׫׮ځ‬ٛ‫ײ׳‬
‫׬‬
‫׬‬
‫׬‬
‫׬׭ځ‬ٛ‫״׫ױ‬
‫װ׬ځ‬ٛ‫׮ױ׭׬‬
‫׬‬
‫׬‬
‫װװ׭‬
‫׫׬׭‬
‫װױ׬‬
‫׫׭׬‬
‫װײ‬
‫׫׮‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
(a)
( What are the ratios
) of consecutive values of s = f(t) ?
f(2) f(3) f(4)
i.e. f(1) , f(2) , f(3) , . . . .
(b) What do these ratios tell us about the value of the iTech device ?
5
question 4
ǓèƐŴNJ 1ƐʟǓŴƐ ٔ )ʂɏɏƐȌɬ ɟňǵƐ ʟňǵʂƐ Ǔɟ ‫׫׫׮ڥ‬ٛ ēňǵʂƐ ʣǓǵǵ žƐŴɏƐňɟƐ ‫׫׮‬%
ɆƐɏ ʬƐňɏٛ
1ʬȌňɟʬɟɬƐȇɟ 1ƐʟǓŴƐ ٔ )ʂɏɏƐȌɬ ɟňǵƐ ʟňǵʂƐ Ǔɟ ‫׫׫׮ڥ‬ٛ ēňǵʂƐ ʣǓǵǵ žƐ‫ځ‬
The table on the left has
a column
giving the value of s = f(t) and
ŴɏƐňɟƐ
$‫ װׯ‬ɆƐɏ ʬƐňɏٛ
well as ∆s.
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׬‬
‫׫׬׭‬
‫ײׯ׬‬
‫׭׫׬‬ٛ‫״‬
‫׭ײ‬ٛ‫׮׫‬
‫׫װ‬ٛ‫׬׭ׯ‬
‫װ׮‬ٛ‫ײׯ״׭‬
‫׫״ځ‬
‫׬‬
‫׮ױځ‬
‫ׯׯځ‬ٛ‫׫׬‬
‫׫׮ځ‬ٛ‫ײ׳‬
‫׬‬
‫׬‬
‫׬‬
‫׬׭ځ‬ٛ‫״׫ױ‬
‫װ׬ځ‬ٛ‫׮ױ׭׬‬
‫׬‬
‫׬‬
‫װװ׭‬
‫׫׬׭‬
‫װױ׬‬
‫׫׭׬‬
‫װײ‬
‫׫׮‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
‫װׯځ‬
(c) What are the ratios of the ∆s and the corresponding initial value
)
(
−63
of s ? i.e. −90
300 , 210 , . . . .
(d) What do these ratios tell us about the value of the iTech device ?
6
exponential functions
Definition
An exponential function f with input variable x is a function of
the form
f(x) = abx
where a and b are real numbers with b > 0.
We call b the base of the exponential function.
We call a the initial value of the exponential function.
7
exponential vs. linear functions
We can characterize exponential functions as follows :
For any two intervals of the input x having the same length, the
percentage change in the output will be the same.
Linear function : Constant rate of change.
Exponential function : Constant percent change.
8
exponential vs. linear functions
For example, let f(x) = 2(1.25)x .
For any interval [x1 , x2 ] having length ℓ :
f(x2 ) − f(x1 )
% change =
= 100
f(x1 )/100
= 100
(
(
2(1.25)x2 − 2(1.25)x1
2(1.25)x1
)
2(1.25)x2
2(1.25)x1
−
2(1.25)x1
2(1.25)x1
)
(
)
= 100 (1.25)x2 −x1 − 1
(
)
= 100 (1.25)ℓ − 1
9
exponential growth and decay
The value of the base determines whether the value of the function
is growing to ±∞ or decaying to 0.
Consider the interval from x = 0 to x = 1 :
% change =
ab − a
ab1 − ab0
=
= 100(b − 1).
0
ab /100
a/100
If b > 1, then the function f(x) = bx is an increasing function. It will
have a positive percentage change over any interval.
If 0 < b < 1, then the function f(x) = bx is a decreasing function. It
will have a negative percentage change over any interval.
10
exponential vs. linear functions
y=
( 1 )x
y = 10x
6
f(x) = bx is
increasing if b > 1,
decreasing if
0 < b < 1.
4
f(x) = bx is positive
for all x.
4
8
y = (.5)x
y = 2x
2
−3
−2
−1
1
−2
2
3
The range of
f(x) = bx is all
positive real
numbers.
vert.-intercept is
(0, 1).
11
question 8
From about 2000 to 2006, home prices in the United States increased
dramatically. Suppose home prices increased by 25% per year.
(a) Complete the following table and graph.
12
question 8
From about 2000 to 2006, home prices in the United States increased
dramatically. Suppose home prices increased by 25% per year.
(b) By what number can we multiply the value of a home at one
moment in time to find its value one year later.
(c) The value of the home at one moment in time is
value one year earlier.
% of its
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