Powerpoint slides copied from or based upon:
Functions Modeling Change
A Preparation for Calculus
Third Edition
Connally,
Hughes-Hallett,
Gleason, Et Al.
Copyright  2007 John Wiley & Sons, Inc.
1
1.5
GEOMETRIC PROPERTIES OF
LINEAR FUNCTIONS
2
Interpreting the Parameters of a
Linear Function
The slope-intercept form for a linear
function is y = b + mx, where b is the
y-intercept and m is the slope.
The parameters b and m can be used
to compare linear functions.
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3
With time, t, in years, the populations of four towns,
PA, PB, PC and PD, are given by the following
formulas:
PA=20,000+1,600t, PB=50,000-300t,
PC=650t+45,000,
PD=15,000(1.07)t
(a) Which populations are represented by linear
functions?
(b) Describe in words what each linear model tells
you about that town's population. Which town starts
out with the most people? Which town is growing
fastest?
Page 35 Example 1
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PA=20,000+1,600t, PB=50,000-300t,
PC=650t+45,000,
PD=15,000(1.07)t
(a) Which populations are represented
by linear functions?
Page 35
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PA=20,000+1,600t, PB=50,000-300t,
PC=650t+45,000,
PD=15,000(1.07)t
(a) Which populations are represented
by linear functions?
Which of the above are of the form:
y = b + mx (or here, P = b + mt)?
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PA=20,000+1,600t, PB=50,000-300t,
PC=650t+45,000,
PD=15,000(1.07)t
(a) Which populations are represented
by linear functions?
Which of the above are of the form:
y = b + mx (or here, P = b + mt)?
A, B and C
Page 35
7
With time, t, in years, the populations of four towns,
PA, PB, PC and PD, are given by the following
formulas:
PA=20,000+1,600t, PB=50,000-300t,
PC=650t+45,000,
PD=15,000(1.07)t
(b) Describe in words what each linear model tells
you about that town's population. Which town starts
out with the most people? Which town is growing
fastest?
Page 35
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PA = 20,000 + 1,600t
b
m
PB = 50,000 + (-300t)
b
PC = 45,000 +
b
when t=0: 50,000
shrinks: 300 / yr
m
650t
when t=0: 45,000
grows: 650 / yr
m
PD = 15,000(1.07)t
Page 35
when t=0: 20,000
grows: 1,600 / yr
when t=0: 15,000
grows: 7% / yr
9
Let y = b + mx. Then the graph of y against x is a
line.
The y-intercept, b, tells us where the line crosses
the y-axis.
If the slope, m, is positive, the line climbs from
left to right. If the slope, m, is negative, the line falls
from left to right.
The slope, m, tells us how fast the line is climbing
or falling.
The larger the magnitude of m (either positive or
negative), the steeper the graph of f.
Page 36 Blue Box
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(a) Graph the three linear functions PA,
PB, PC from Example 1 and show how
to identify the values of b and m from
the graph.
(b) Graph PD from Example 1 and
explain how the graph shows PD is not
a linear function.
Page 36 Example 2
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Note vertical intercepts. Which are climbing?
Which are climbing the fastest? Why?
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A
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13
Intersection of Two Lines
To find the point at which two lines intersect,
notice that the (x, y)-coordinates of such a point
must satisfy the equations for both lines. Thus, in
order to find the point of intersection
algebraically, solve the equations simultaneously.
If linear functions are modeling real quantities,
their points of intersection often have practical
significance. Consider the next example.
Page 37
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The cost in dollars of renting a car for a day from
three different rental agencies and driving it d
miles is given by the following functions:
C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d
(a) Describe in words the daily rental
arrangements made by each of these three
agencies.
(b) Which agency is cheapest?
Page 37 Example 3
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C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d
(a) Describe in words the daily rental
arrangements made by each of these three
agencies.
C1 charges $50 plus $0.10 per mile driven.
C2 charges $30 plus $0.20 per mile.
C3 charges $0.50 per mile driven.
Page 37
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What are the points of intersection?
A
C1
C2
Page
C3
17
C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d
Page 37
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C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d
Page 37
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C1 = 50 + 0.10d, C2 = 30 + 0.20d, C3 = 0.50d
Page 37
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A
C1
C2
Page 38
C3
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Thus, agency 3 is cheapest up to 100 miles.
A
C1
C2
Page 38
C3
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Agency 1 is cheapest for more than 200 miles.
A
C1
C2
Page 38
C3
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Agency 2 is cheapest between 100 and 200 miles.
A
C1
C2
Page 38
C3
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Equations of Horizontal and Vertical Lines
An increasing linear function has positive
slope and a decreasing linear function has
negative slope.
What about a line with slope m = 0?
Page 38
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Equations of Horizontal and Vertical Lines
An increasing linear function has positive
slope and a decreasing linear function has
negative slope.
What about a line with slope m = 0? If the rate
of change of a quantity is zero, then the
quantity does not change.
Thus, if the slope of a line is zero, the value of
y must be constant. Such a line is horizontal.
Page 38
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Plot the points:
(-2, 4)
(-1,4)
(1,4)
(2,4)
Page 38
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The equation y = 4
represents
a linear
A
function with slope
m = 0.
Page 38
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Plot the points:
(-2, 4)
(-1,4)
(1,4)
(2,4)
Let's verify by using any two pointsLet's use (-2,4) & (1,4):
m?
Page 38
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Plot the points:
(-2, 4)
(-1,4)
(1,4)
(2,4)
Let's verify by using any two pointsLet's use (-2,4) & (1,4):
y
m
?
x
Page 38
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Plot the points:
(-2, 4)
(-1,4)
(1,4)
(2,4)
Let's verify by using any two pointsLet's use (-2,4) & (1,4):
y y1  y0 4  4 0
m


 0
x x1  x0 1  2 3
Page 38
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Now plot the points:
(4,-2)
(4,-1)
(4,1)
(4,2)
Page 38
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A
Page 38
The equation x = 4 does not
represent a linear function
with slope m = undefined.
33
Now plot the points:
(4,-2)
(4,-1)
(4,1)
(4,2)
Let's verify by using any two pointsLet's use (4,-2) & (4,1):
m?
Page 38
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Now plot the points:
(4,-2)
(4,-1)
(4,1)
(4,2)
Let's verify by using any two pointsLet's use (4,-2) & (4,1):
y
m
?
x
Page 38
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Now plot the points:
(4,-2)
(4,-1)
(4,1)
(4,2)
Let's verify by using any two pointsLet's use (4,-2) & (4,1):
y y1  y0 1  2 3
m


  undefined!
x x1  x0
44 0
Page 38
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For any constant k:
The graph of the equation y = k is a horizontal
line and its slope is zero.
The graph of the equation x = k is a vertical
line and its slope is undefined.
Page 39 Blue Box
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Slope of Parallel (||) & Perpendicular ( ) lines:
|| lines have = slopes, while  lines have
slopes which are the negative reciprocals of
each other.
So if l1 & l2 are 2 lines having slopes m1 & m2,
respectively. Then:
these lines are || iff m1 = m2
these lines are  iff m1 = - 1 / m2
Page 39 Blue Box
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So if l1 & l2 are 2 lines having slopes m1 & m2,
respectively. Then:
these lines are || iff m1 = m2
these lines are iff m1 = - 1 / m2
Page 39
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Are these 2 lines || or
 or neither?
y = 4+2x, y = -3+2x
Page N/A
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Are these 2 lines || or
 or neither?
y = 4+2x, y = -3+2x
Parallel (m = 2 for both)
Page N/A
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Are these 2 lines || or
 or neither?
y = -3+2x, y = -.1 -(1/2)x
Page N/A
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Are these 2 lines || or
 or neither?
y = -3+2x, y = -.1 -(1/2)x
: m1 = 2, m2 = -(1/2)
Page N/A
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End of Section 1.5.
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