Download sec 1.3 - Cerritos College

Copyright © 2007 Pearson Education, Inc.
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Chapter 1: Linear Functions, Equations,
and Inequalities
1.1 Real Numbers and the Rectangular Coordinate
System
1.2 Introduction to Relations and Functions
1.3 Linear Functions
1.4 Equations of Lines and Linear Models
1.5 Linear Equations and Inequalities
1.6 Applications of Linear Functions
Copyright © 2007 Pearson Education, Inc.
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1.3 Linear Functions
• Linear Function f ( x)  ax  b
– a and b are real numbers
– Its graph is called a line
– Its solution is an ordered pair, (x,y), that makes the equation true
Example
f ( x)  3 x  6
The points (0,6) and (–1,3) are solutions of y  3x  6
since 6 = 3(0) + 6 and 3 = 3(–1) + 6.
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1.3 Graphing a Line Using Points
• Graphing the line y  3x  6
Connect with a straight line.
x
y
2
0
1
3
0
6
1
9
Plot the ordered pairs
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1.3 Graphing a Line with the TI-83
• Graph the line y  3x  6 with the TI-83
Xmin=-10, Xmax=10, Xscl=1
Ymin=-10, Ymax=10,Yscl=1
Copyright © 2007 Pearson Education, Inc.
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1.3 The x- and y-Intercepts, Zero of a
Function
• x-intercept: let y = 0 and solve for x
• y-intercept: let x = 0 and solve for y
• Zero of a function is any number c where f(c) = 0
• Two distinct points determine a line
- e.g. (0,6) and (–2,0) are the y- and x-intercepts of the
line y = 3x + 6, and x = –2 is the zero of the function.
Copyright © 2007 Pearson Education, Inc.
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1.3 Graphing a Line Using the Intercepts
Example: Graph the line y  2 x  5 .
x
y
x-intercept
0
5
y-intercept
2.5
0
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1.3 Application of Linear Functions
A 100 gallon tank is initially full of water and being drained at a rate of
5 gallons per minute.
a) What is the linear function that models this problem?
f ( x )  (constant rate of change) x  initial amount
f ( x )  5 x  100
b) How much water is in the tank after 4 minutes?
f ( 4)  55( 4)  100  80 gallons
c) Interpret the x- and y-intercepts.
y-intercept, let x  0  y  5(0) 100 100
meaning that the tank initially has 100 gallons in it.
x-intercept, let y 0  05x 100 x 20 minutes
meaning that the tank takes 20 minutes to empty.
Copyright © 2007 Pearson Education, Inc.
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1.3 Constant Function
• Constant Function f ( x)  b
–
–
–
–
–
–
b is a real number
the graph is a horizontal line
y-intercept: (0,b)
domain  (, )
range  {b}
Example: f ( x)  3
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Slide 1-9
1.3 Graphing with the TI-83
• Different views with the TI-83
f ( x)  3x  6
f ( x)  3x  6
• Comprehensive graph shows all intercepts
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1.3 Slope
• Slope of a Line
In 1984, the average annual cost
for tuition and fees at private fouryear colleges was $5991. By 2004,
this cost had increased to $20,082.
The line graphed to the right is
actually somewhat misleading,
since it indicates that the increase
in cost was the same from year to
year.
$20, 082  $5991 $14, 091

 $705.
2004  1984
20
The average yearly cost was $705.
Copyright © 2007 Pearson Education, Inc.
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1.3 Formula for Slope
• Slope m
y
(x , y )
2
2
y  y
(x , y )
1
1
x  x
2
 x
2
 y
1
(x , y )
2 1
1
x
0
y y2  y1
m

x x2  x1
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1.3 Example: Finding Slope Given Points
Determine the slope of a line passing through
points (2, 1) and (5, 3).
y2  y1 3  (1) 4
4
m



x2  x1  5  2  7
7
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1.3 Graph a Line Using Slope and a Point
• Example using the slope and a point to graph a
line
– Graph the line that passes through (2,1) with slope 
y
(2,1)
(2,1)
x
0
y
y
(2,1)
x
0
(5,-3)
down 4
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4
3
x
0
(5,-3)
right 3
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1.3 Slope of Horizontal and Vertical Lines
• Slope of ay horizontal line is 0
(1,4)
(0,4)
x
0
44 0
m
 0
1 0 4
• Slope of a vertical line
y
(4,4)
0
4
x
40 4
m
  undefined
44 0
• Equation of a vertical line that passes through the
point (a,b):
xa
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1.3 Slope-Intercept Form of a Line
Slope-intercept form of the equation of a line
f ( x )  ax  b
or
y  ax  b
– m  a is the slope, and
– b is the y-intercept
Copyright © 2007 Pearson Education, Inc.
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1.3 Matching Examples
1. y  2 x  3
A.
2. y  2 x  3
B.
3. y  2 x  3
C.
Solution: 1) C, 2) A, 3)B
Copyright © 2007 Pearson Education, Inc.
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1.3 Application of Slope
•
Interpreting Slope
–
In 1980, passengers traveled a total of 4.5 billion miles on
Amtrak, and in 2000 they traveled 5.5 billion miles.
a)
Find the slope m of the line passing through the points (1980,
4.5) and (2000, 5.5).
Solution:
b)
m  5.5  4.5  1  0.05
2000 1980 20
Interpret the slope.
Solution: Average number of miles people are
traveling on Amtrak increased by around .05
billion, or 50 million miles per year.
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