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04/21/23 Math 120 - KM 1
Chapter 3
3.1 Cartesian Coordinates
• 3.2 Linear Functions: Graphs and Slope
• 3.3 More on Graphing Linear Equations
• 3.4 Finding Equations of Lines; Applications
04/21/23 Math 120 - KM 2
The Cartesian PlaneRene Descartes (1596 - 1650)
2.1
04/21/23 Math 120 - KM 3
Coordinates of Points
2.1
04/21/23 Math 120 - KM 4
Plotting Points
2.1
04/21/23 Math 120 - KM 5
Plot: y = 2x - 1
x y
0 -1
1/2 0
1 1
10 ,
0
2
1,
11,
2.1
04/21/23 Math 120 - KM 6
Plot: y = -2x
x y
0 0
0 0
1 -2
2 -4
21,
00,
42 ,
2.1
04/21/23 Math 120 - KM 7
2.2 Linear Functions: Graphs and
Slope
2.4
04/21/23 Math 120 - KM 8
Linear Function: f(x) = mx + b
Remember y is the same as f(x).
f(x) = mx + b or
y = mx + b
(x, y)(x, f(x))
2.4
04/21/23 Math 120 - KM 9
Linear Function: f(x) = mx + b
What does b do?
y = x
y = x + 3
y = x - 4
(0,3)
(0,0)
(0,-4)
2.4
04/21/23 Math 120 - KM 10
Linear Function: f(x) = mx + b
What does m do?
y = x
y = 3xy = -4x
(1,3)
(1,1)
(1,-4)
2.4
04/21/23 Math 120 - KM 11
y = mx+b(0, b) is the y-intercept
m is the slope
y = mx + b
(0,b)RUN
1
RISEm
2.4
04/21/23 Math 120 - KM 12
Equation Slope y-intercept
y = 2x - 5
y = -x + 1
y = x
y = 3
x = 2
3x- 2y - 4 = 0
y = mx+bm is the slope
(0, b) is the y-intercept
(0,-5)
(0, 1)
(0,0)
(0,3)
no y-intercept
m = 2
m = -1
m = 1
m = 0
no slope
(0,-2)m = 3/2
2.4
04/21/23 Math 120 - KM 13
SLOPEBASICS
POSITIVE SLOPE The line rises from left to right.
ZERO SLOPE The line is HORIZONTAL (zero rise)
NEGATIVE SLOPEThe line falls from left to right.
UNDEFINED SLOPEThe line is VERTICAL (zero run)
2.4
04/21/23 Math 120 - KM 14
Computing the Slope:(x1,y1) and (x2, y2)
are coordinates of two points
Slope
xy
21
12
12 xx,xxyy
m
xinchange
yinchange
RunRise
xdelta
ydelta
0Run
2.4
04/21/23 Math 120 - KM 15
A Slope Triangle
21
12
12 xx,xxyy
m
P1(x1, y1)
RUNx2 - x1
RISEy2 - y1
P2(x2, y2)
RISE
RUN
2.4
04/21/23 Math 120 - KM 16
Compute the Slope
(-2, -3)
(4, -1)
3
1
6
2
24
31
)(
m
Upwards + Slope
2.4
04/21/23 Math 120 - KM 17
Another Slope
(-4, -3)
(0, 5)
21
2
4
8
40
35
m
Upw
ards
+
Slo
pe
2.4
04/21/23 Math 120 - KM 18
Compute this Slope
(-5, 2)
(4, -1)
3
1
9
3
54
21
)(
m
DOWNWARDS - Slope
2.4
04/21/23 Math 120 - KM 19
Compute the Slope
(-5, -1)
(4, -1)
0
9
0
54
11
)(
m
HORIZONTALO
2.4
04/21/23 Math 120 - KM 20
Compute the Slope
(-4, -5)
(-4, 0)
Undefined
)(m
0
5
44
50
VER
TIC
AL
no s
lope
2.4
04/21/23 Math 120 - KM 21
Practice Problem 1a
3
2
Determine the slope of the line
containing points P1 and P2. ),(P 571 ),(P 912
6
4
71
59
)(
m
RUN = 3
RISE = 2
The line rises from left to right.
2.4
04/21/23 Math 120 - KM 22
Practice Problem 1b
Determine the slope of the line
containing points P1 and P2. ),(P 531 ),(P 232
0
7
33
52
)(
m
The slope is undefined.
Vertical Line
2.4
04/21/23 Math 120 - KM 23
Practice Problem 1c
2
1
Determine the slope of the line
containing points P1 and P2. ),(P 301 ),(P 542
4
2
04
35
)(
m
RUN = 2
RISE = -1
The line falls from left to right.
2.4
04/21/23 Math 120 - KM 24
Practice Problem 1d
0
Determine the slope of the line
containing points P1 and P2. ),(P 571 ),(P 512
6
0
71
55
)(
m
RUN = 6RISE = 0
The line is Horizontal.
2.4
04/21/23 Math 120 - KM 25
2.5 More on Graphing Linear Equations
2.5
04/21/23 Math 120 - KM 26
Intercepts
x-intercept ( , 0)
Y – intercept (0 , )
2.5
04/21/23 Math 120 - KM 27
Sketch the Graph
x-intercept ( , 0)
Y – intercept (0 , )
(3, 0)
(0, 6)
2.5
04/21/23 Math 120 - KM 28
Sketch the Graph
x-intercept ( , 0)
Y – intercept (0 , )
(1, 0)
(1, 3)
2.5
04/21/23 Math 120 - KM 29
Sketch the Graph
x-intercept ( , 0)
Y – intercept (0 , )
(3, 0)
(0, -2)
2.5
04/21/23 Math 120 - KM 30
Sketch the Graph
x-intercept ( , 0)
Y – intercept (0 , )
(3, 2)(0, 2)
2.5
04/21/23 Math 120 - KM 31
Sketch the Graph
x-intercept ( , 0)
Y – intercept (0 , )
(3, 2)
(0, 0)
2.5
04/21/23 Math 120 - KM 32
Graph a Line using a Point and the
Slope
P1(x1, y1)
P2(x1+ RUN, y1+ RISE)
RUN
RISE
2.5
04/21/23 Math 120 - KM 33
Graph
(0, - 4) is the y-intercept
2/5 is the slopeRun 5 then rise 2
(0+5, - 4+2) or (5, -2) is the second point.
45
2 xy
(0, -4)
(5, -2)
2.5
04/21/23 Math 120 - KM 34
Graph
(0, 4)
(2, 3)
42
1 xy
(0, 4) is the y-intercept
-1/2 is the slopeRun 2 then rise -1
(0+2, 4+-1) or (2, 3) is the second point.
2.5
04/21/23 Math 120 - KM 35
Graph
(0, 0)
(3, -2)
032 yx
xy3
2Solve for y
(0, 0) is the y-intercept
-2/3 is the slopeRun 3 then rise -2
(0 + 3, 0 + -2) or (3, -2) is the second point.
2.5
04/21/23 Math 120 - KM 36
Uh oh, no y !
(-4, 0)
082 x4x
The x-coordinate is always - 4,so this is a vertical line! Plot (-4,
#)
2.5
04/21/23 Math 120 - KM 37
Uh Oh, no x?
(0, 2)
1882 y
2yThe y-coordinate is always 2,
so this is a horizontal line! Plot (#, 2)
2.5
04/21/23 Math 120 - KM 38
Special Pairs of Lines
2.5
04/21/23 Math 120 - KM 39
Parallel Lines
Are in the same plane And
never intersect!
Parallel Lines have the same
slope!
2.5
04/21/23 Math 120 - KM 40
Perpendicular Lines
Intersect to form a right angle
(90 degrees)!
Perpendicular lines have slopes that are
negative reciprocalsThe slopes multiply to -1.(as long as they are not vertical or horizontal.)
2.5
04/21/23 Math 120 - KM 41
m m|| m
What’s My Slope?
73
2 xy
3
2
3
2
2
3
73
2 xy
2.5
04/21/23 Math 120 - KM 42
m m|| m
What’s My Slope?1243 yx
4
3
3
4
4
3
34
3 xy
2.5
04/21/23 Math 120 - KM 43
m m|| m
What’s My Slope?0 yx
1 11
xy 1
2.5
04/21/23 Math 120 - KM 44
m m|| m
What’s My Slope?
14y
0 UND0
14y
2.5
04/21/23 Math 120 - KM 45
m m|| m
What’s My Slope?
03 x
UND 0UND
03 x
2.5
04/21/23 Math 120 - KM 46
Solve this Mystery!
632 yx
Are these lines parallel
764 yx
23
2 xy
6
7
3
2 xy
2.5
04/21/23 Math 120 - KM 47
Solve this Mystery!
632 yx
Are these lines parallel
1246 yx
23
2 xy
32
3 xy
2.5
04/21/23 Math 120 - KM 48
OK … here it comes!
Determine the equation of the line through the point (4,1) that is perpendicular
to y = -3x+7
3m3
1m
)x()(y 43
11
2.5
04/21/23 Math 120 - KM 49
2.6 Finding Equations of Lines;
Applications
2.6
04/21/23 Math 120 - KM 50
Linear Equations
CByAx
bmxy
)xx(myy 11
2.6
04/21/23 Math 120 - KM 51
What’s my line?
bmxy
)xx(myy 11
3
2m ),(P 15
Clues!
)x()(y 53
21
2.6
04/21/23 Math 120 - KM 52
Look Closely!
bmxy
)xx(myy 11
4
3m ),(P 50
Aha!
54
3
xy
2.6
04/21/23 Math 120 - KM 53
Where’s m ?
12
12
xxyy
m
)xx(myy 11
),(P 861
Uh oh!
),(P 742
)( 64
87
2
1
))(x(y 62
18
2.6
04/21/23 Math 120 - KM 54
Look !
),(P 1731 ),(P 1742
17y
2.6
04/21/23 Math 120 - KM 55
I bet I can’t fool you !
),(P 1731 ),(P 032
3x
2.6
04/21/23 Math 120 - KM 56
A Chemistry Problem?
In the Fahrenheit system, water freezes at 32oF
and boils at 212oF.In the Celsius system, water
freezes at 0oC and boils at 100oC.
Using (F, C) , (32o , 0o), and (212o, 100o)
determine a formula to calculate C if F is known!
2.6
04/21/23 Math 120 - KM 57
A Chemistry Problem - clues?
(F, C) (32o , 0o) (212o, 100o)
)xx(myy 11
)FF(mCC 11
9
5
180
100
32212
0100
m
2.6
04/21/23 Math 120 - KM 58
A Chemistry Problem - clues?
(F, C) (32o , 0o) (212o, 100o)
)FF(mCC 11
(F1, C1)
)F(C 329
50
9
160
9
5 FC
2.6
04/21/23 Math 120 - KM 59
Up in the Air?
An ultralight crop duster is climbing at a rate of 800
ft/min.
Write an equation which gives the height of the
plane in terms of the time after take off.
What is the height 3 minutes after take off?
2.6
04/21/23 Math 120 - KM 60
Up in the Air – Clues!
An ultralight crop duster is climbing at a rate of 800
ft/min.
2.6
04/21/23 Math 120 - KM 61
Up in the Air – Clues!
The crop duster takes off from ground level. So his height at
the start is 0 feet.
2.6
04/21/23 Math 120 - KM 62
Up in the Air – Clues!
What is the height 3 minutes after take off? t = 3
2.6
04/21/23 Math 120 - KM 63
That’s All For Now!
04/21/23 Math 120 - KM 64
Where We Left Off Last Class
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