1.5

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1.5   Geometric Properties of Linear Functions
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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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With time, t, in years, the populations of four
towns, PA, PB, PC and PD, are given by the
following formulas PA20,0001,600t,
PB50,000-300t, PC650t45,000,
PD15,000(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?
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PA20,0001,600t, PB50,000-300t,
PC650t45,000, PD15,000(1.07)t (a) Which
populations are represented by linear functions?
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PA20,0001,600t, PB50,000-300t,
PC650t45,000, PD15,000(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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PA20,0001,600t, PB50,000-300t,
PC650t45,000, PD15,000(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
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With time, t, in years, the populations of four
towns, PA, PB, PC and PD, are given by the
following formulas PA20,0001,600t,
PB50,000-300t, PC650t45,000,
PD15,000(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?
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PA 20,000 1,600t when t0 20,000

grows 1,600 / yr b
m PB 50,000 (-300t) when t0
50,000
shrinks 300 / yr b
m PC 45,000 650t when t0
45,000
grows 650 / yr b
m PD 15,000(1.07)t when t0
15,000
grows 7 / yr
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• Let y b mx. Then the graph of y against x is
  a line.
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• 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.

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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.
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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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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.
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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?
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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.
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What are the points of intersection?
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C1
C2
C3
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C1 50 0.10d, C2 30 0.20d, C3 0.50d
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C1 50 0.10d, C2 30 0.20d, C3 0.50d
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C1 50 0.10d, C2 30 0.20d, C3 0.50d
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A
C1
C2
C3
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Thus, agency 3 is cheapest up to 100 miles.
A
C1
C2
C3
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Agency 1 is cheapest for more than 200 miles.
A
C1
C2
C3
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Agency 2 is cheapest between 100 and 200 miles.
A
C1
C2
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?
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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.
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Plot the points (-2, 4) (-1,4) (1,4) (2,4)
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The equation y  4 represents a linear function
with slope m  0.
A
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Plot the points (-2, 4) (-1,4) (1,4)
(2,4) Let's verify by using any two
points- Let's use (-2,4) (1,4)
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Plot the points (-2, 4) (-1,4) (1,4)
(2,4) Let's verify by using any two
points- Let's use (-2,4) (1,4)
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Plot the points (-2, 4) (-1,4) (1,4)
(2,4) Let's verify by using any two
points- Let's use (-2,4) (1,4)
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Now plot the points (4,-2) (4,-1) (4,1) (4,2)
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The equation x  4 does not represent a linear
function with slope m  undefined.
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Now plot the points (4,-2) (4,-1) (4,1) (4,2)
Let's verify by using any two points- Let's
use (4,-2) (4,1)
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Now plot the points (4,-2) (4,-1) (4,1) (4,2)
Let's verify by using any two points- Let's
use (4,-2) (4,1)
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Now plot the points (4,-2) (4,-1) (4,1) (4,2)
Let's verify by using any two points- Let's
use (4,-2) (4,1)
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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.
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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
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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
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Are these 2 lines or or neither?   y
42x, y -32x
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Are these 2 lines or or neither?   y
42x, y -32x Parallel (m 2 for both)
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Are these 2 lines or or neither?   y
-32x, y -.1 -(1/2)x
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Are these 2 lines or or neither?   y
-32x, y -.1 -(1/2)x m1 2, m2 -(1/2)
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End of Section 1.5.