Problems on Linear and Piecewise Linear Functions
Two variables, say xand y, are said to be linearly related if and only if there are
constants mand bso that y=mx +b.Thusifxand yare related by the equation
4x−2y= 8 then they are linearly related as it is possible to solve for yto get y=2x−4
(in this case m= 2 and b=−4). Likewise if the variables Aand Rare related by the
equation 5A−3R=7thenAand Rare linearly related as it is possible to solve for A
and get A=5
3R+7
3(whence this time m=5
3and b=7
3).
The basic most basic fact about a linear relation is that its graph is a straight line.
As we all know the slope of this line is the “rise” (i.e. the change in the yvalue) divided
by the “run” (the change in the xvalue).
r
r
(x0,y
0
)
(x, y)
run =x−x0=∆x
rise =y−y0=∆y
m=slope =rise
run =y−y0
x−x0
The slope mis the same regardless of which pair of points (x, y)and(x
0
,y
0
)areused.
Thus if we think of (x0,y
0
) as a fixed point and (x, y ) as a variable point then the change
in xis ∆x=x−x0and the change in yis ∆y=y−y0.Then∆xand ∆yare also
variables and the equation of the line can be written as
∆y=m∆xwhere ∆x=x−x0,∆y=y−y0.
This is the same as y−y0=m(x−x0) which is what is often called the point slope form
of the line.
Here are some problems to review what you know about linear functions and to intro-
duce you to piecewise linear functions.
1. If a car is moving at constant velocity, then the relation between distance, D,and
time, t, is linear. For the rest of this problem it will be assumed that a car is moving
a constant velocity along a straight road running by campus, that after one hour it
is 10 miles from campus and after three hours it is 50 miles from campus.
(a) What is the speed of the car in miles per hour?
(b) Give a formula for the distance D.Let∆t=t−1and∆D=D−10 then
write this equation in the form ∆D=m∆t.
(c) Using your answer to last part graph Das a function of t.
(d) Use your equation to predict when the distance of the car from campus when
t= 5 hours.
(e) What is the time when the car is at a distance of 170 miles from campus?
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