Download Section 1.4

1.4
Equations of Lines and Modeling
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D E T E R M I N E E Q U AT I O N S O F L I N E S .
G I V E N T H E E Q U AT I O N S O F T W O L I N E S , D E T E R M I N E
W H E T H E R T H E I R G R A P H S A R E PA R A L L E L O R
PERPENDICULAR.
M O D E L A S E T O F D ATA W I T H A L I N E A R F U N C T I O N .
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Addison Wesley
Slope-Intercept Equation
Recall the slope-intercept equation y = mx + b or
f (x) = mx + b.
If we know the slope and the y-intercept of a line, we
can find an equation of the line using the slopeintercept equation.
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Example
A line has slope  7 and y-intercept (0, 16). Find an
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equation of the line.
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Publishing as Addison Wesley
Example
2
A line has slope  and contains the point (–3, 6). Find
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an equation of the line.
Using the point (3, 6), we substitute –3 for x and 6 for
y, then solve for b.
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Publishing as Addison Wesley
Point-Slope Equation
The point-slope equation of the line with slope m
passing through (x1, y1) is
y  y1 = m(x  x1).
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Example
Find the equation of the line containing the points
(2, 3) and (1, 4).
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Parallel Lines
Vertical lines are parallel. Non-vertical lines are
parallel if and only if they have the same slope and
different y-intercepts.
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Perpendicular Lines
Two lines with slopes m1 and m2 are perpendicular if
and only if the product of their slopes is 1:
m1m2 = 1 or
m2 = -1/m1
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Addison Wesley
Perpendicular Lines
Lines are also perpendicular if one is vertical (x = a)
and the other is horizontal (y = b).
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as Addison Wesley
Example
Determine whether each of the following pairs of lines is
parallel, perpendicular, or neither.
a) y + 2 = 5x,
5y + x = 15
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Example (continued)
Determine whether each of the following pairs of lines is
parallel, perpendicular, or neither.
b)
2y + 4x = 8,
5 + 2x = –y
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Example (continued)
Determine whether each of the following pairs of lines is
parallel, perpendicular, or neither.
c) 2x +1 = y,
y + 3x = 4
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Example
Write equations of the lines (a) parallel and
(b) perpendicular to the graph of the line 4y – x = 20
and containing the point (2, 3).
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Publishing as Addison Wesley
Mathematical Modeling
When a real-world problem can be described in a
mathematical language, we have a mathematical model.
The mathematical model gives results that allow one to
predict what will happen in that real-world situation. If
the predictions are inaccurate or the results of
experimentation do not conform to the model, the model
must be changed or discarded. Mathematical modeling
can be an ongoing process.
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Publishing as Addison Wesley
Curve Fitting
In general, we try to find a function that fits, as well as
possible, observations (data), theoretical reasoning, and
common sense. We call this curve fitting, it is one aspect
of mathematical modeling.
In this chapter, we will explore linear relationships.
Let’s examine some data and related graphs, or scatter
plots and determine whether a linear function seems to
fit the data.
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Example
The gross domestic product (GDP) of a country is the market
value of final goods and services produced. Market value depends
on the quantity of goods and services and their price. Model the
data in the table below on the U.S. Gross Domestic Product with a
linear function. Then estimate the GDP in 2012.
Find the line of best fit for the scatterplot.
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Modeling Linear Regression
Credit-Card Debt. Model the data given in the table below with a
linear function, and estimate the average credit-card debt per U.S.
household in 2005 and in 2014. Use 1990 as the year the beginning
of the model.
Year
CC Debt
1992
$3,803.00
1996
$6,912.00
2000
$8,308.00
2004
$9,577.00
2008
$10,691.00
Copyright © 2012 Pearson Education,
Inc. Publishing as Addison Wesley
f(x) = 411.025x + 3746.95