Module
3
3.1
Linear Functions
Functions of the form
are called linear
functions because their graphs are straight lines. The number m
gives the slope (rise/run) of the line
and b is the value of f(x) at x =
0 (the y-intercept of
the graph).
In your reading of Section 2.3, you reviewed various ways of determining lines from their equations and vice versa. We shall now look at that material as applied to linear functions.
Q1
Finding
a Linear Function from Information About Its Graph
Any geometric description of a line determines a linear function (the one whose graph is the given line). For example, since a straight line is determined by just two points, we should be able to determine a linear function given any two points on its graph. To check that you can do this, find the formula for the linear function f having f(1)=2 and f(2)=1. (see answer#1)
An Applied Example
Note that in the situation described in the introduction, it says that “for each $5 she raises her price she loses 4 sales.…..” . So if she raises the price by $10, which is like two $5 increases one after the other, she will lose a total of 8 sales, and so on. On the other hand, if she lowers the price her sales should go up, and for smaller price changes we should get proportionately smaller changes in number sold. For example, if she raises the price by just $1 we can expect the sales loss to be just 1/5 of the 4, that is 0.8. (You might worry about the number of sales not being a whole number, but since we are talking about the average number of sales per week, it might indeed be a fraction - e.g. if she sells 24 one week and 25 the next, then the average for those two weeks is 24.5 sales per week).
Q2
Now, what if she charges a
price of $p per pair? In that case, the number of $5
increases would
be (p-50)/5, so the average number N of sales per week
would be given by
, or equivalently
.
If you graph this relation you will get a straight line of slope -4/5
with
at p
= 0. So the N
intercept
is 90. Where is the p-intercept? What
does the p-intercept tell us about
the shoe store? (see answer #2)
Q3
The description of the problem
can be thought
of as specifying the line
in point-slope form. What is the given
point,
and how is the slope specified?
(see answer #3).
Further Practice re Lines and Linear Functions
Check your understanding, and practice for speed, by working through some of the Exercises on p132-7. As a minimum, try #3,23,33,43,53.
Answers to Questions re Linear Functions
1. There are various ways to do this.
One way is to write the given
conditions as
two equations in two unknowns, using the facts that
to get
,
and then to solve by
elimination or
substitution to get
But it may be quicker to use
the two point
form
for the equation of the line (see
Module 1.4),
and just re-arrange it to get
,
so
.
2. The p-intercept
(where
) is at
= 112.5 . This is the price at which
all
potential customers would go elsewhere
i.e. no-one would buy the shoes at the
store.
Note: If we
try to set a price above this, then the formula gives a negative value
for N . This “negative number of sales”
doesn’t really make sense (at a stretch you might interpret it as
meaning that
people would come in and try to sell you their own shoes but that’s
probably
not realistic), so we should exclude values of p > 112.5 .
Negative prices are probably also unrealistic
(unless we are willing to pay people to take our shoes), so the domain
of the
sales function should be restricted to
3. The first point given was (p,N)=(50,50) and the slope was specified by the statement “for each $5 she raises her price she loses 4 sales”.