Growth
Authors: Bill Davis, Horacio Porta and Jerry Uhl
©19962007
Publisher: Math Everywhere, Inc.
Version 6.0
1.01 Growth
BASICS
B.1) Growth of line functions
f
@
x
D
=
a x
+
b
A line function
f
@
x
D
is any function whose formula is
f
@
x
D
=
a x
+
b
where
a
and
b
are constants.
Here's a plot of a line function:
Clear
@
f, x
D
;
a
=
0.5;
b
=
4;
f
@
x
_
D
=
a x
+
b;
Plot
@
f
@
x
D
,
8
x,

2, 8
<
,
PlotStyle
>
88
Red, Thickness
@
0.015
D<<
,
AxesLabel
>
8
"x"
,
"f
@
x
D
"
<
, PlotLabel
>
"A line function"
D

2
2
4
6
8
x
4
5
6
7
8
f
@
x
D
A line function
There's steady growth as
x
advances from left to right.
Here's another:
Clear
@
f, x
D
;
a
= 
0.3;
b
=
2;
f
@
x
_
D
=
a x
+
b;
Plot
@
f
@
x
D
,
8
x,

2, 8
<
,
PlotStyle
>
88
Red, Thickness
@
0.015
D<<
,
AxesLabel
>
8
"x"
,
"f
@
x
D
"
<
, PlotLabel
>
"A line function"
D

2
2
4
6
8
x
0.5
1.0
1.5
2.0
2.5
f
@
x
D
A line function
Steady (negative) growth as
x
advances from left to right.
Play with other choices of
a
and
b
until you get the feel of a line function.
·
B.1.a.i) Constant growth rate
The most important feature of a line function
f
@
x
D
=
a x
+
b
is revealed by the following calculation.
Clear
@
f, a, b, x, h
D
;
f
@
x
_
D
=
a x
+
b;
Expand
@
f
@
x
+
h
D

f
@
x
DD
a h
What feature of line functions is revealed by this calculation?
·
Answer:
The calculation reveals that when you take a line function
f
@
x
D
=
a x
+
b
,
then you find that
f
@
x
+
h
D

f
@
x
D
=
a h
.
This tells you that when
x
advances by
h
units, then
f
@
x
D
grows by
a h
units.
Consequently a line function
f
@
x
D
=
a x
+
b
has constant growth rate of
a
units on the
f
@
x
D
axis for each unit on the
x
axis.
·
B.1.a.ii) The meaning of the growth rate.
As you saw above, the growth rate of a line function
f
@
x
D
=
a x
+
b
measures out to
a
units
on the
f
@
x
D
axis per unit on the
x
axis.
What is the significance of
a
?
·
Answer:
Big positive
a
's force bigtime fast growth as the following true scale plot shows:
Clear
@
f, x
D
;
a
=
8;
b
=
2;
f
@
x
_
D
=
a x
+
b;
bigpositive
=
Plot
@
f
@
x
D
,
8
x,

1, 2
<
,
PlotStyle
>
88
Red, Thickness
@
0.015
D<<
,
PlotLabel
>
"Big a
>
0"
, AspectRatio
>
Automatic
D
1.0
1.5
2.0

5
5
10
15
Big a
>
0
Small positive
a
's force slow growth as the following true scale plot shows:
Clear
@
f, x
D
;
a
=
0.2;
b
=
2;
f
@
x
_
D
=
a x
+
b;
smallpositive
=
Plot
@
f
@
x
D
,
8
x,

1, 5
<
,
PlotStyle
>
88
Red, Thickness
@
0.015
D<<
,
PlotLabel
>
"Small a
>
0"
, AspectRatio
>
Automatic
D

1
1
2
3
4
5
2.0
2.2
2.4
2.6
2.8
3.0
Small a
>
0
a
=
0
forces no growth at all as the following true scale plot shows:
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@
f, x
D
;
a
=
0;
b
=
2;
f
@
x
_
D
=
a x
+
b;
smallpositive
=
Plot
@
f
@
x
D
,
8
x,

1, 5
<
,
PlotStyle
>
88
Red, Thickness
@
0.015
D<<
,
AxesLabel
>
8
"x"
,
"f
@
x
D
"
<
,
PlotLabel
>
"a
=
0"
, AspectRatio
>
Automatic
D

1
1
2
3
4
5
x
1
2
3
4
f
@
x
D
a
=
0
Small negative
a
's force slow (negative) growth as the following true scale plot shows:
Clear
@
f, x
D
;
a
= 
0.2;
b
=
2;
f
@
x
_
D
=
a x
+
b;
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 Spring '08
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 Exponential Function, Exponentiation, Exponential decay

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