Understanding Linear Functions
Understanding Linear Functions
In this component we see how the basic properties of linear
functions relate to data gathered experimentally. There are three
sections:
We begin by seeing how rates of change define linear functions.
Linear functions are the functions that appear most often when we
apply mathematics. The key property of linear functions is that
they have a constant rate of change. To see what this means, let's
look at a simple example.
The following table shows the height, in centimeters, of a
sunflower in terms of its age in days.
| Age (days) | Height (centimeters) |
|
|
| 60 | 210 |
| 61 | 212 |
| 62 | 214 |
| 63 | 216 |
| 64 | 218 |
| 65 | 220 |
The pattern is obvious: The sunflower grows 2 centimeters every
day. In mathematical terms, this says that the height has a
constant rate of change with respect to the age. That rate of
change is 2 centimeters per day. This means that the height is a
linear function of the age. Once we have observed this pattern, we
can easily make predictions about the height of the sunflower at
other times. For example, we can find the height at age 68 days
as follows:
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Height atage 65 + Three days of growth |
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So far we have seen an example of a linear function-a function
with a constant rate of change. For a linear function that rate
of change is usually called the slope. Thus the slope of
the linear function in the example above is 2 centimeters per
day. You should note that the units of the slope always work out
this way, namely as the units of the function divided by the units
of the variable.
In the example above it was easy to see the pattern. But what if
the height is measured every 2 days instead of every day? Let's
look at an example like that. Here is a table showing the height
of a different plant.
| Age (days) | Height (centimeters) |
|
|
| 60 | 208 |
| 62 | 213 |
| 64 | 218 |
| 66 | 223 |
| 68 | 228 |
| 70 | 233 |
It's not hard to see the pattern here. The plant grows 5
centimeters every 2 days. That means that it grows
5/2 = 2.5 centimeters per day. This height function has
a constant rate of change, so it is a linear function. The slope
is 2.5 centimeters per day.
These examples suggest how to test data to see if they represent a
linear function. Suppose that we have a table of data in which
the values of the variable are equally spaced. (This means that
we have measurements at constant intervals-for example, every
two days.) Then a linear function represents the data when the
values of the function are also equally spaced, and in this case
the slope is the ratio of these two spacings:
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Slope = |
Change in function
Change invariable
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. |
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This basic formula summarizes almost all of what we
need to know about linear functions! Note that it can be
rearranged as
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Change in function = Slope×Change in variable. |
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The basic formula can also be
rearranged as
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Change in variable = |
Change in function
Slope
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. |
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We will use all of these forms.
Let's test our understanding by looking at a couple of different
examples. Here is a table showing the number, in thousands, of
bacteria present as a function of time in hours.
| Time (hours) | Number of bacteria (thousands)
|
|
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| 0 | 3 |
| 2 | 12 |
| 4 | 48 |
| 6 | 192 |
Is there a linear function representing the data? Because the
population is measured every 2 hours, the values of the variable
are equally spaced. Are the function values equally spaced? Over
the first two hours the population grows by 12-3 = 9 thousand
bacteria, an absolute growth rate of 9/2 = 4.5 thousand
bacteria per hour. Over the next two hours the population grows by
48-12 = 36 thousand bacteria, an absolute growth rate of
36/5 = 18 thousand bacteria per hour. Already we can
see that no linear function represents the data: The function
values are not equally spaced, and the rate of change (the growth
rate) is not constant. In fact, checking the last two-hour
interval in the table shows a growth of 192-48 = 144 thousand
bacteria, an absolute growth rate of 144/2 = 72
thousand bacteria per hour. This is a still different number! Of
course we should not be surprised that the growth rate isn't
constant. The most natural model for population growth is
exponential, not linear. (As we will see later, however, many
functions can be approximated by a linear function over a short
interval.) You may have already noticed that the population in the
table doubles every hour. This is a good reminder to use common
sense when analyzing data.
For another example, let's look at the speed of sound in air,
which is usually modeled as a linear function of the temperature.
Here is a partial table showing the speed of sound, in meters per
second, as a function of the temperature in degrees Celsius.
| Temperature (degrees Celsius) | Speed of sound (meters per second)
|
|
|
| 10 | 337.4 |
| 16 | |
| 20 | 343.4 |
| 346.4 |
Given that the speed is a linear function of the temperature, how
can we fill in the blanks in the table? First we need to find the
slope of this linear function by computing the rate of change from
the complete pairs of data points that we know. We will use the
basic formula stated above:
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Slope = |
Change in function
Change in variable
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. |
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When the temperature
increases from 10 to 20 degrees Celsius, the speed increases
by 343.4-337.4 = 6 meters per second. Thus the slope is
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Change in function
Change invariable
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Hence the slope is 0.6 meter per second
per degree Celsius.
Using the slope we can fill in the first blank in the table. Here
we need the rearrangement of the basic formula:
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Change in function = Slope×Change in variable. |
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Going from a temperature of 10 degrees to a
temperature of 16 degrees represents a change of 6 degrees,
and so the speed of sound increases by
Thus
the change is 3.6 meters per second. We add this result to the
speed at a temperature of 10 degrees to get the answer
337.4+3.6 = 341 meters per second for the speed of sound at a
temperature of 16 degrees Celsius.
To fill in the second blank in the table we use another
rearrangement of the basic formula:
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Change in variable = |
Change in function
Slope
|
. |
|
Using
the last two lines of the table, we have a change of
346.4-343.4 = 3 meters per second in the function, and we want to
know the change in the variable. Because the slope is 0.6 meter
per second per degree Celsius, the change in the variable is
Thus the change in the variable is 5 degrees
Celsius, so the temperature for the last line of the table is
20+5 = 25 degrees Celsius.
The formula for a linear function is simple and easy to use, and
we'll see now how to find that formula based on a table of data.
Let's consider again the partial table showing the speed of sound,
in meters per second, as a function of the temperature in degrees
Celsius. For simplicity, let's drop the lines having blank
entries:
| Temperature (degrees Celsius) | Speed of sound (meters per second)
|
|
|
| 10 | 337.4 |
| 20 | 343.4 |
We let T denote the temperature, in degrees Celsius, and S the
speed of sound in meters per second. Then S is a linear
function of T. The first step in finding a formula for S is
to determine the slope of the linear function. Earlier we found
this to be 0.6 meter per second per degree Celsius. The second
step in finding a formula can be visualized if we add a new line
to the table:
| Temperature (degrees Celsius) | Speed of sound (meters per second)
|
|
|
| 10 | 337.4 |
| 20 | 343.4 |
| T | S |
If we consider the change from the first line to the last line, we
see that the change in the variable is T-10 and that the change
in the function is S-337.4. Now we use the rearrangement of the
basic formula that we used before:
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Change in function = Slope×Change in variable. |
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Thus we
have
where we have used the fact that the
slope is 0.6. This is the formula for the linear function. Of
course, it can be written as
or even as
The graph is shown below.
This example is typical of what we do to find a linear formula
from a table exhibiting linear data. Suppose that the variable is
x and the function is y. First we find the slope using any
pair of data points. Let's call the slope m. Then we use any
line in the table-say, x = x1 and y = y1-in the formula
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Change in function = Slope×Change in variable |
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to get
This is called the
point-slope form of the linear function. This can be rearranged
to give
with b = y1-mx1. Here b is the initial
value of the linear function and also the vertical intercept of
the line that is its graph. This form of the equation is called
the slope-intercept form. Thus in the example above the
point-slope form is
and the slope-intercept
form is
We have seen that a table of data in which the values of the
variable are equally spaced is represented by a linear function if
the values of the function are equally spaced. In this case the
slope is the ratio of these two spacings:
|
Slope = |
Change in function
Change in variable
|
. |
|
The formula for such a linear function (say for y
in terms of x) is
if m is the slope and
(x1,y1) is any entry in the table.
Often a formula is valid only for certain choices of the variable.
For example, the height of a sunflower may be appropriately
modeled by a linear function over the middle portion of the time
when it's growing, but a linear model might not be appropriate
over the early stages of growth. This leads to the question of
how to model data that are not exactly linear but may have some
similarities to linear data. That question will be discussed in
the section on modeling.
To prepare for the section on modeling and review this one,
consider the following example of data adapted from a study of
enzyme kinetics. Here the variable s is the substrate (which has
the units of mM NPG, millimolar di-nitro-phenol) and the function
v is absorbance (which has units of OD, optical density) per
second.
| s = Substrate | v = Absorbance per second
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| 0.5 | 0.0030 |
| 1.0 | 0.0066 |
| 1.5 | 0.0102 |
| 2.0 | 0.0138 |
Show that the data are represented by a linear function, find a
formula for that function, and draw its graph. Determine whether
the initial value makes sense physically. If not, determine what
this says about the validity of a linear model over all choices of
substrate.
File translated from TEX by TTH, version 2.25.
On 08 Aug 2002, 14:44.