Download Chap 7

44
VAN WYK’S MATH 304
7. Linear Functions and Lines (Last updated 5/20/11)
7.1. Functions. We will now shift our focus from the set of integers, Z, to the set of real
numbers, R. Although you should be familiar with the notion of a real-valued function, we’ll
run through a quick review.
A relation on R is a set of ordered pairs (x, y) where both x (the first coordinate) and y
(the second coordinate) are real numbers. Such a relation is a function if no two ordered
pairs have the same first coordinate. If f can be represented by an equation, we write
y = f (x), and say y is a function of x. The set of all values of x is the domain of f . The
variable x is known as the independent variable, while y is known as the dependent
variable31.
Example 7.1.
(1) The table below defines a function f :
x
f (x)
−5 − 12
7
3 101 53
0
7
0 − 11
√
2
7
The domain of f is dom(f) = {−5, − 12 , 0, 3, 101 53 }. Notice f (−5) = f (101 35 ) = 7,
so there are two ordered pairs with the same second coordinate.
√
defines a function, since every “allowable” value of x
(2) The equation f (x) = x+3
x
completely determines the value of y. The domain32 of f is the set of all real values
√
of x for which the expression x+3
is defined. Since square roots are only defined for
x
non-negative values, we must have x + 3 ≥ 0, so x ≥ −3. And since division by zero
is undefined, we must have x $= 0. Putting these two constraints together, we have
the domain of f is the set of all reals greater than or equal to −3, except for 0; in
symbols dom(f) = {x | −3 ≤ x < 0 or x > 0}.
(3) The graph in Figure 1 defines y as a function of x. Notice that the graph passes the
“vertical line test” – every vertical line intersects the graph in at most one point –
which guarantees that no two ordered pairs (x, y) have the same first coordinate.
Definition 7.1. If y = f (x) with x0 and x1 contained in dom(f), where x0 < x1 , then the
(x0 )
∆y
= f (xx11)−f
.
average rate of change in f between x = x0 and x = x1 is ∆x
−x0
Example 7.2. Using the function in Example 7.1 (3), the average rate of change in f
(−1.4)
between x = −1.4 and x = 3.2 is f (3.2)−f
≈ 0.4−2.4
≈ −0.44. Notice that the average
3.2−(−1.4)
4.6
rate of change only depends on the two points (−1.4, 2.4) and (3.2, 0.4), not on the behavior
of the function between those two points.
31The
32The
basic idea is that the value of x determines the value of y, but not necessarily vice-versa.
domain of a function defined by an equation is sometimes referred to as the natural domain.
FUNDAMENTALS OF ALGEBRA
45

















Figure 1. A function defined by a graph.
7.2. Linear Functions. We are interested in functions whose rate of change is the same
over all intervals. In other words, we want
what x0 and x1 are.
f (x1 )−f (x0 )
x1 −x0
= m for some constant m, no matter
Since we want this equation to hold for any x0 and x1 , we can take x1 = x > 0 and
(0)
= m, or, equivalently, f (x) = mx + f (0).
x0 = 0. The above equation then becomes f (x)−f
x−0
On the other hand, if we take x1 = 0 and x0 = x < 0, then the above equation becomes
f (0)−f (x)
0−x
= m, which is also equivalent to f (x) = mx + f (0).
Since f (0) is the y-intercept33 of the graph of f , we’ll set f (0) = b. Then the constraint
that the average rate of change of f equals the constant m forces f to be of the form
f (x) = mx + b.
Definition 7.2. A linear function is a function that can be written in the form f (x) =
mx + b.
∆y
By construction, the graph of a linear function is a straight line with slope m = ∆x
and
y-intercept b. Since the average rate of change is constant, it can be computed using any
two points on the graph of f .
Example 7.3. Suppose we have the following information about the function f :
x −2 0 2
f (x)
4
2 1 0 −2
0−2
Then the average rate of change of f between x = −2 and x = 2 is 2−(−2)
= − 21 , while its
−2−2
= − 23 , so f is not linear.
average rate of change between x = −2 and x = 4 is 4−(−2)
33The
y-intercept is the y-value corresponding to the value x = 0, so it is always f (0).
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VAN WYK’S MATH 304
7.3. Lines. In the equation f (x) = mx + b, both the slope and the y-intercept are apparent.
This is one of the standard forms of a line. It and others are listed below:
Slope-intercept form. y = mx + b, where m is the slope and b is the y-intercept.
Point-slope form. y − y0 = m(x − x0 ), where m is the slope and (x0 , y0 ) is any
point on the line.
Two point form. y − y0 =
points on the line.
y1 −y0
(x
x1 −x0
− x0 ), where (x0 , y0 ) and (x1 , y1 ) are any two
Standard form. Ax + By = C.
Notice the names of the first three forms above indicate which would be most appropriate
to use, given the information you have or the information you want. The next example
illustrates this.
Example 7.4.
(1) If we know the slope of a line is −3 and the y-intercept is 72 , then it
would be easiest to use the slope-intercept form for the line: y = −3x + 72 .
(2) If we know the slope of a line is −3 and a point on the line is (4, −9), then it would
be easiest to write its point-slope form: y + 9 = −3(x − 4). However, since there
are infinitely many points on the line, there are infinitely many possible point-slope
forms, so we might want the slope-intercept form, which is unique. This can be done
by solving the point-slope equation for y: y + 9 = −3x + 12, so y = −3x + 3.
(3) If we know the line passes through the points (4, −9) and (−5, 18), then it would
(x − 4) or y − 18 =
be easiest to write one of its two-point forms: y − 9 = 18−(−9)
−5−4
−9−18
(x
4−(−5)
+ 5). Solving either of these equations for y yields the slope-intercept form
y = −3x + 3.
It appears the standard form of the line, Ax + By = C, is somewhat out of place, since it
is not apparent that the constants A, B, or C have any graphical interpretation, and because
none of those constants are unique: if Ax + By = C, then (kA)x + (kB)y = kC for any k.
But, in fact, standard form is the only form that every line possesses, including vertical lines.
Since any two points on a vertical line have the same x-coordinate, it is not a function – the
vertical line test fails miserably – and its slope does not exist34 because ∆x = 0. However,
its standard form is x = x0 , where x0 is the x-coordinate of any (and hence all) points on
the line.
34How
can you explain to a child that division by zero is undefined? One way is to realize division as
repeated subtraction. For example, 72 = 3 21 because 2 can be subtracted from 7 exactly 3 21 times. In this
model, 07 would represent how many times 0 can be subtracted from 7. Another way is to realize division as
sharing equally. For example, 72 = 3 12 because if 7 objects are shared equally between 2 people, then each
person gets 3 21 objects. In this model, 70 would represent how many objects each of 0 people would get if
7 objects were shared equally. Lastly, we can consider fact families. For example, if 07 = a, then 7 = a · 0,
which is impossible.
FUNDAMENTALS OF ALGEBRA
47
!!!
!
C # !!
!!
C !!
!!!
!
!
!
!
!!A#
B#
!!
!!
!
!
!
! A
B
!!
!
!
Figure 2. Any two points suffice for computing the slope of a line.
Also, since a line is a degenerate case of a conic section35, which has general form Ax2 +
Bxy + Cy 2 + Dx + Ey + F = 0, the standard form of a line puts it in this context.
Finally, there is actually a geometric interpretation of the constants A, B, and C, which
we will see at the end of this section.
The relationship between standard form and slope-intercept form is explored in Problem 4.
∆y
, is the average rate of change in
7.4. Slope. As we saw above, the slope of a line, m = ∆x
y with respect to x between any two points on the line. The slope of a line captures only
this rate of change, and tells us nothing about the location of the line in the plane; a point
is required to do that.
Other than the derivation of linear function above, one can use the notion of similar
triangles to justify that any two points may be used to determine the slope of a line. The
right triangles ∆ABC and ∆A# B # C # in Figure 2 are similar, since they have equal angles, so
the ratios of their sides are equal. In particular, the ratio of their vertical to horizontal sides
are equal, and this is the slope of the line !.
Loosely, the slope of a line indicates how far from a horizontal line it is. A horizontal line
has a slope of zero, lines which increase (from left to right) have positive slopes, and lines
which decrease have negative slopes. The steeper the line (either increasing or decreasing),
the greater the absolute value of the slope.
Two facts about slope you are probably familiar with are:
(1) Parallel lines have the same slope.
(2) Perpendicular36 lines have slopes that are negative reciprocals37 of each other.
35The
conic sections include parabolas, ellipses, hyperbolas, and circles; each can be realized as the
intersection of a pair of cones with a plane.
36Two lines are perpendicular if the angle between them is 90 degrees.
37Two numbers a and b are negative reciprocals of each other if ab = −1. Thus, for example, 3 and − 7
7
3
are negative reciprocals.
48
VAN WYK’S MATH 304
y
y = − αβ x $
$
#
$
$
$
$!
(−β, α) $
$
y = αβ x
!!
!
!
!!!
$
$
!! (α, β)
$
!!
"
$!!
!!$
!!
$
!!
$
!!
$
!!
$
!!
$
$
$
$
$
$
$
x
Figure 3. Slopes of perpendicular lines.
While the first of these is clear, the second is not. To illustrate the second fact, consider
rotating the coordinate axes, which are perpendicular by construction, about the origin by
some acute angle, as in Figure 3. The resulting two lines are then also perpendicular, since
the angle between the axes is preserved during the rotation. If (α, β) is a point on one of
the lines, then the congruent rectangles show that (−β, α) is a point on the other. Notice
that the slopes of the two lines are negative reciprocals of each other.
Each pair of perpendicular lines can be realized by translating the point of intersection of
two such lines above to other points in the plane.
7.5. Higher dimensions. There is a geometric interpretation of lines in the plane that
generalizes nicely to higher dimensions. The point-slope form of a line illustrates that a
point and a slope (direction) determines a line. Rather than using the slope to determine
the direction of the line, we could use a vector38 that is perpendicular to the line. Thus a
line can be described as the locus of points in the plane that contains a given point and is
perpendicular to a given vector. In fact, this is one benefit of the standard form of a line:
the line Ax + By = C is perpendicular to the vector $n = (A, B). See Figure 4.
Moving to 3-space, the locus of points in space that contains a given point and is perpendicular to a given vector is a plane rather than a line. In fact, the equation of a plane
38A
vector is a quantity with both magnitude and direction.
FUNDAMENTALS OF ALGEBRA
49
y
#
Ax + By = C
$n = (A, B)
%%
%%
!"
%%
!
%%
!
%%
!
%%!!
%%
%%
%%
"
%%
%%
%%
x
Figure 4. Points on a line normal to $n in the plane.
z #
Ax + By
(
(
(
(
(
()
( )
) $
n = (A, B, C)
)
$
) #
+ Cz = D
#
)
#
! )
(
)
(
)
"
(
&
!!
(
&
!!
!
( & !!
( &!!
(&
!!
(!
&
&
'
&
y
x
Figure 5. Points on a line normal to $n in three space.
in standard form is Ax + By + Cz = D, and this plane is perpendicular39 to the vector
$n = (A, B, C). See Figure 5.
39The
standard term for perpendicular here is “normal,” which is why this vector is denoted !n.
50
VAN WYK’S MATH 304
So the locus of points that contains a given point and is perpendicular to a given vector
in the 2-dimensional plane is a 1-dimensional line, while the locus of points that contains a
given point and is perpendicular to a given vector in 3-dimensional space is a 2-dimensional
plane. Similarly, in all higher dimensions, the locus of points that contains a given point
and is perpendicular to a given vector is a hyperplane of dimension one less than the space
in which it sits.
Problems.
1. Draw the graphs of three different functions that have the same average rate of change
between x = −3 and x = 3 as the function f (x) = 2x + 1.
2. Find the average rate of change:
(a) of the function f (x) = x2 − 4 between x = 0 and x = 2, and between x = −2 and
x = 2.
(b) of the function f (x) defined by the table below, between x = 0 and x = 4, and
between x = −4 and x = 4.
x −4 −2
f (x)
0 2 4
7
0 − 11
7
2
7
7
(c) of the function graphed below between x = −5 and x = 4 (approximately).













3. If ! is the line that passes through the points (0, −4) and (−2, 5), write an equation for !
in the following forms:
(a) point-slope form
(b) slope-intercept form
(c) standard form, where A, B, and C are integers.
FUNDAMENTALS OF ALGEBRA
51
A
4. Show the line with standard form Ax + By = C with B $= 0 has slope − B
and y-intercept
C
. Hint: Use slope-intercept form.
B
What type of line is it if B = 0?
5. Label each of the following lines with the correct slope: −2, − 12 , 0, 12 , 2, and none. Identify
the pairs of perpendicular lines.
$
$
$
$
*
*
$
$
*
*
*
*
%%
$
!!
*
!
%%
!
$
*
%%
$
!!
*
%%
!!
$
*
%% $ * !!
%%
$!
* !
!
%
! *$ %
!! * $ %%
!
%%
!
$
*
!
%%
!
$
*
!
%%
!
$
*
!
%%
!
$
*
$
*
$
*
$
*
$
*
$
*
$
*
6. Let ! be the line passing through the points (−2, 3) and (5, −4). Find the slope-intercept
form of the following lines.
(a) The line parallel to ! that passes through the point (1, 2).
(b) The line perpendicular to ! that passes through the point (1, 2).
7. Find equations for and sketch graphs of two lines parallel to the line x = 3 and two lines
perpendicular to the line x = 3.
8. Consider an arithmetic sequence with first term a and common difference d: a, a + d, a +
2d, a + 3d, . . . . This sequence determines points in the plane:
x 1
2
3
4
...
y a a + d a + 2d a + 3d . . .
Show that each of these points (x, y) lies on the line y = dx + (a − d). (In other words,
there is a linear relationship between the subscripts of an arithmetic sequence (an ) and
the terms of the sequence.)
52
VAN WYK’S MATH 304
9. Consider a geometric sequence with first term a and common ratio r: a, ar, ar2 , ar 3 , . . . .
This sequence determines points in the plane:
x 1
2
3
4
...
y a ar ar 2 ar 3 . . .
Show that there is not a linear relationship between the subscripts of a geometric sequence
(an ) and the terms of the sequence.
10. Verify that the line 3x − 4y = 1 is perpendicular to the vector $n = (3, −4).
Hint: Find the slope of any line parallel to $n.
11. The life expectancy of males on Isla de Muerta in the year 2000 was 70.3 years, while in the
year 2010 it was 72.1 years. Find a linear function E(t) that fits these data points, where
t is the number of years since 2000, and use it to approximate what the life expectancy
of males will be in the year 2025. (Round all numbers to the nearest tenth.)
12. Geneva decided to take a bike ride on one long street in her neighborhood. Below is
a graph that gives her distance (in miles) from her house versus the elapsed time (in
minutes) she was riding.
d [mi]
#
(60, 3)
,
,
,
,
,
,
,
,
!(90, 6)
!
,
*+
,
* + ,
*
+!, (75, 2)
*
(30, 1)
!!
!
!
!!
!
!
(0, 0)
*
!*(45, 1)
"
t [min]
(a) Describe her ride, including how fast she was riding (in miles per hour), how long she
was riding at that pace, and whether she was riding away from or toward her house.
(b) Find her average velocity over her entire ride (in miles per hour), that is, find the
average rate of change in the distance she was from her house between t = 0 minutes
and t = 90 minutes.