linear functions and precalculus, algebra 2 mathematcs. Fundamentals

2.1 Linear
Functions
Represent a linear
function
Determine whether a
linear function is
increasing, decreasing,
or constant
Calculate and interpret
slope
Write the point-slope
form of an equation
Write and interpret a
linear function

2.1 Linear
Functions
• The train’s distance from the
station is a function of the time
during which the train moves at a
constant speed plus its original
distance from the station when it
began moving at constant speed

Can the input in
the previous
example be any
real number?
THE INPUT REPRESENTS TIME,
SO WHILE NONNEGATIVE
RATIONAL AND IRRATIONAL
NUMBERS ARE POSSIBLE,
NEGATIVE REAL NUMBERS ARE
NOT POSSIBLE FOR THIS
EXAMPLE
THE INPUT CONSISTS OF NON-
NEGATIVE REAL NUMBERS
ANOTHER WAY TO REPRESENT
LINEAR FUNCTIONS IS
VISUALLY, USING A GRAPH
THE DOMAIN IS COMPRISED
OF ALL REAL NUMBERS
BECAUSE ANY NUMBER MAY
BE DOUBLED, AND THEN HAVE
ONE ADDED TO THE PRODUCT

Linear
Function
A linear function is a function
whose graph is a line
Linear functions can be written
in the slope-intercept form of a
line where is the initial or
starting value of the function ,
and is the constant rate of
change, or slope of the function
The y-intercept is at

Using a Linear Function to
Find the Pressure on a
Diver THE PRESSURE, IN POUNDS PER
SQUARE INCH ON THE DIVER IN
FIGURE 4 DEPENDS UPON HER
DEPTH BELOW THE WATER
SURFACE, IN FEET
THIS RELATIONSHIP MAY BE
MODELED BY THE EQUATION,
RESTATE THIS FUNCTION IN WORDS

Solution
• To restate the function in words,
we need to describe each part of
the equation
• The pressure as a function of
depth equals four hundred thirty-
four thousandths times depth
plus fourteen and six hundred
ninety-six thousandths

Analysis
The initial value, 14.696, is the
pressure in PSI on the diver at a
depth of 0 feet, which is the
surface of the water
This tells us that the pressure on
the diver increases 0.434 PSI for
each foot her depth increases
The linear functions we used in
the two previous examples
increased over time, but not
every linear function does
A linear function may be
increasing, decreasing, or
constant

Increasing and
Decreasing Functions

Deciding
whether a
Function Is
Increasing,
Decreasing,
or Constant
• ⓐ The total number of texts a
teen sends is considered a
function of time in days
• ⓑ A person has a limit of 500
texts per month in their data plan
• ⓒ A person has an unlimited
number of texts in their data plan
for a cost of $50 per month

Solution
• ⓐ The function can be
represented as where is the
number of days
• ⓑ The function can be
represented as where is the
number of days
• ⓒ The cost function can be
represented as because the
number of days does not affect
the total cost

Are the units for slope always ?
Think of the units as the change of output value for
each unit of change in input value
An example of slope could be miles per hour or
dollars per day
Notice the units appear as a ratio of units for the
output per units for the input

Calculate Slope
• The slope, or rate of change,
of a function can be
calculated according to the
following: where and are
input values, and are
output values

How To
Given two points from
a linear function,
calculate and interpret
the slope
Determine the units for
output and input
values
Calculate the change of
output values and
change of input values
Interpret the slope as
the change in output
values per unit of the
input value

Example 3
If is a linear
function, and and
are points on the
line, find the
slope
Is this function
increasing or
decreasing?

Solution
The coordinate pairs are
and To find the rate of
change, we divide the
change in output by the
change in input
We could also write the
slope as The function is
increasing because

Analysis
• As noted earlier, the order in
which we write the points
does not matter when we
compute the slope of the
line as long as the first
output value, or y-
coordinate, used
corresponds with the first
input value, or x-coordinate,
used

Try It #1
If is a linear
function, and and
are points on the
line, find the
slope
Is this function
increasing or
decreasing?

Finding the
Population
Change from
a Linear
Function
• The population of a city increased
from 23,400 to 27,800 between
2008 and
• Find the change of population per
year if we assume the change was
constant from 2008 to

Solution
• The rate of change relates the
change in population to the
change in time
• The population increased by
people over the four-year time
interval
• To find the rate of change, divide
the change in the number of
people by the number of years

Analysis
Because we are told
that the population
increased, we would
expect the slope to be
positive
This positive slope we
calculated is therefore
reasonable

Try It #2
The population of a small
town increased from
1,442 to 1,868 between
2009 and
Up until now, we have
been using the slope-
intercept form of a linear
equation to describe
linear functions
The point-slope form is
derived from the slope
formula

Point-Slope Form of a Linear Equation
The point-slope form of a
linear equation takes the
form where is the slope,
are the coordinates of a
specific point through
which the line passes
The point-slope form is
particularly useful if we
know one point and the
slope of a line
If we wanted to then
rewrite the equation in
slope-intercept form, we
apply algebraic techniques

Example 5
• Write the point-slope form
of an equation of a line with
a slope of 3 that passes
through the point Then
rewrite it in the slope-
intercept form

Solution
Let’s figure out what we know from the given information
The slope is 3, so We also know one point, so we know and Now we
can substitute these values into the general point-slope equation
Then we use algebra to find the slope-intercept form

Try It #3
Write the point-slope form of an equation of a line with a slope of
that passes through the point Then rewrite it in the slope-intercept
form
The point-slope form of an equation is also useful if we know any two
points through which a line passes
Both equations describe the line shown in Figure

Writing Linear Equations Using Two Points
Write the point-slope
form of an equation of
a line that passes
through the points
and
Then rewrite it in the
slope-intercept form

Solution
So Next, we substitute the slope and the coordinates for one of the
points into the general point-slope equation
We can choose either point, but we will use The point-slope equation
of the line is To rewrite the equation in slope-intercept form, we use
algebra
The slope-intercept equation of the line is

Try It #4
Write the point-slope form of an
equation of a line that passes
through the points and Then
rewrite it in the slope-intercept form
Now that we have written equations
for linear functions in both the
slope-intercept form and the point-
slope form, we can choose which
method to use based on the
information we are given
If we want to rewrite the equation
in the slope-intercept form, we
would find If we wanted to find the
slope-intercept form without first
writing the point-slope form, we
could have recognized that the line
crosses the y-axis when the output
value is

How To
Given the graph of a
linear function, write an
equation to represent
the function
Identify two points on
the line
Use the two points to
calculate the slope
Determine where the
line crosses the y-axis to
identify the y-intercept
by visual inspection
Substitute the slope and
y-intercept into the
slope-intercept form of a
line equation

Writing an Equation for a
Linear Function
Write an equation for a linear function given a
graph of shown in Figure

Solution Identify two points on
the line, such as and
Use the points to
calculate the slope
the coordinates of one
of the points into the
point-slope form
We can use algebra to
the slope-intercept form

Analysis
• This makes sense because we can
see from Figure 11 that the line
crosses the y-axis at the point ,
which is the y-intercept, so

Writing an
Equation for a
Linear Cost
Function
SUPPOSE BEN STARTS A COMPANY
IN WHICH HE INCURS A FIXED COST
OF $1,250 PER MONTH FOR THE
OVERHEAD, WHICH INCLUDES HIS
OFFICE RENT
HIS PRODUCTION COSTS ARE
$37.50 PER ITEM
WRITE A LINEAR FUNCTION
WHERE IS THE COST FOR ITEMS
PRODUCED IN A GIVEN MONTH

Solution
The fixed cost is present every month, $1
The costs that can vary include the cost to produce each item, which
is $37.50 for Ben
The variable cost, called the marginal cost, is represented by The cost
Ben incurs is the sum of these two costs, represented by

Analysis
• If Ben produces 100 items in a
month, his monthly cost is
represented by So his monthly
cost would be $5

Writing an Equation for a Linear
Function Given Two Points
If is a linear function, with , and , find an equation
for the function in slope-intercept form

Solution We can write the given
points using coordinates
the coordinates of one
of the points into the
point-slope form
We can use algebra to
the slope-intercept form

Try It #5
If is a linear function, with
and find an equation for the
function in slope-intercept
form
In the real world, problems
are not always explicitly
stated in terms of a function
or represented with a graph
Fortunately, we can analyze
the problem by first
representing it as a linear
function and then
interpreting the components
of the function

How To
Given a linear function and the initial value and rate of change, evaluate
Determine the initial value and the rate of change
Substitute the values into
Evaluate the function at

Example 10
• Using a Linear Function to
Determine the Number of Songs
in a Music Collection Marcus
currently has 200 songs in his
music collection
• Every month, he adds 15 new
songs
• Write a formula for the number of
songs, in his collection as a
function of time, the number of
months

Solution
The initial value for this function
is 200 because he currently
owns 200 songs, so which
means that The number of
songs increases by 15 songs per
month, so the rate of change is
15 songs per month
Therefore we know that We can
substitute the initial value and
the rate of change into the
slope-intercept form of a line
We can write the formula With
this formula, we can then
predict how many songs Marcus
will have in 1 year

Analysis
• Notice that N is an increasing
linear function
• As the input increases, the output
increases as well

Using a Linear Function to
Calculate Salary Plus
Commission
WORKING AS AN INSURANCE
SALESPERSON, ILYA EARNS A BASE
SALARY PLUS A COMMISSION ON
EACH NEW POLICY
THEREFORE, ILYA’S WEEKLY
INCOME, DEPENDS ON THE
NUMBER OF NEW POLICIES, HE
SELLS DURING THE WEEK
LAST WEEK HE SOLD 3 NEW
POLICIES, AND EARNED $760 FOR
THE WEEK

Solution
• The value of is the starting value
for the function and represents
Ilya’s income when or when no
new policies are sold
• We can interpret this as Ilya’s
base salary for the week, which
does not depend upon the
number of policies sold
• Our final interpretation is that
Ilya’s base salary is $520 per week
and he earns an additional $80
commission for each policy sold

Example 12
• Using Tabular Form to Write an
Equation for a Linear Function
Table 1 relates the number of rats
in a population to time, in weeks
• Use the table to write a linear
equation

Solution
Is the initial value always provided in a table of
values like Table 1?

Try It #6
• A new plant food was introduced
to a young tree to test its effect
on the height of the tree
• Table 2 shows the height of the
tree, in feet, months since the
measurements began
• Write a linear function, where is
the number of months since the
start of the experiment

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