Download Section 3.1

Chapter 10
Graphing Equations and Inequalities
1
Section 10.1
Rectangular Coordinates
2
Rectangular Coordinate System
Also called Cartesian, after René Descartes
y
x
Ordered Pair (3, 4); the point where x = 3 and y = 4
3
Example
• Plot the following: (-2, 4), (-1, -1), (3, 4), (1, -3)
4
Solution
• Plot the following: (-2, 4), (-1, -1), (3, 4), (1, -3)
5
Identify the Points
1
1
6
Solution
• (4,3)
• (-3, 5)
• (-4, -3)
• (-1, -4)
• (3, - 4)
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Draw a Scatter Diagram
Year
2003
2004
2005
2006
2007
2008
2009
2010
Sales, $M
3
3.4
3.8
4.2
4.6
5
5.4
5.8
8
Solution
Sales, $M
3
3.4
3.8
4.2
4.6
5
5.4
5.8
7
6
5
Sales, $M
Year
2003
2004
2005
2006
2007
2008
2009
2010
4
3
2
1
0
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011
Year
9
Draw a Scatter Graph
Year
2003
2004
2005
2006
2007
2008
2009
Tornadoes
1376
1817
1264
1106
1098
1691
1156
10
Solution
Year
2003
2004
2005
2006
2007
2008
2009
Tornadoes
1376
1817
1264
1106
1098
1691
1156
2000
Number of Tornadoes
1800
1600
1400
1200
1000
800
2002
2003
2004
2005
2006
Year
2007
2008
2009
2010
11
Why Make a Graph?
• Spot trends
– Are your grades improving?
– How is the stock market doing?
– Are we really experiencing global warming?
• Can compare two lines and find differences
• Allows us to transfer information quickly – we understand
pictures better than we understand lists of numbers
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Standard Form of Grid; Terminology
13
Give the quadrant of the following coordinates
• (3, 4)
• (-2, 4)
• (2, -5)
• (3, -5)
• (-1, -1)
14
Solution
• (3, 4) Q1
• (-2, 4) Q2
• (2, -5) Q4
• (3, -5) Q1
• (-1, -1) Q3
15
Ordered Pairs are Solutions to an Equation
• 2x + y = 8 or y = 8 – 2x
– The coordinate pair (0, 8) or x = 0, y = 8 is a solution
– The coordinate pair (1, 8) or x = 1, y = 8 is not a solution
16
Example
• y = 12 – 3x
x
-2
-1
0
1
2
3
y = 12 - 3x
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Solution
• y = 12 – 3x
x
-2
-1
0
1
2
3
y = 12 - 3x
18
15
12
9
6
3
18
Example
• y = 3x
x
-2
-1
0
1
2
3
y = 3x
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Solution
• y = 3x
x
-2
-1
0
1
2
3
y = 3x
-6
-3
0
3
6
9
20
Example
• y = (1/2) x - 5
x
-2
-1
0
1
2
3
y = 1/2 x - 5
21
Solution
• y = (1/2) x - 5
x
-2
-1
0
1
2
3
y = 1/2 x - 5
-6
-5.5
-5
-4.5
-4
-3.5
22
Section 10.2
Graphing Linear Equations
23
Linear Equations
• ay = bx + c
• a, b and c are any numbers
• A graph of a linear equation in two variables results in a
straight line when graphed in a coordinate plane
• Examples:
y = 4x – 3
9y = -4x + ½
y/2 = -3/8 x + 5000
• The terms can be on different sides of the equal sign:
x = 5y + 2; 3 = x + y
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Are these linear equations?
• y = 3x – 4
• x=4
• y = 2(x - 3) + 2
• x = 5y – 3(y + 2)
• y = 3x2 – x + 5
• x = y + 21
25
Graphing
• Make a table of at least three values
• Plot points
• Join with line
26
Example
• y = 12 – 3x
27
Solution
y = 12 - 3x
18
15
12
9
6
3
20
16
12
8
y
x
-2
-1
0
1
2
3
4
0
-4
-2
-4
0
2
4
x
28
• y = 3x
29
y = 3x
4
3
y
-3
0
3
2
1
0
y
x
-1
0
1
-2
-1
-1 0
1
2
-2
-3
-4
x
30
• Y = 1/2 x - 5
31
y = 1/2 x - 5
-6
-5.5
-5
-4.5
-4
-3.5
2
0
-4
y
x
-2
-1
0
1
2
3
-2
0
2
4
-2
-4
-6
x
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• y = -2
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• y = -2
x
y
-1
0
1
2
-2
-2
-2
-2
1
-2
-1
0
1
2
3
-1
-3
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Section 6.3
Intercepts and Special Lines
35
Intercepts
• An intercept is the value where a line crosses an axis
x intercept is the value where a line crosses the x axis
y intercept is the value where a line crosses the y axis
• We write these values as coordinate pairs:
x intercept is (x, 0), since the x axis is at y = 0
y intercept is (0,y), since the y axis is at x = 0
36
Find the x Intercept
• 3x – y = 4
• 2x + 5y + 1 = 0
• x=¾y+5
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Solution x intercept
• 3x – y = 4
y=0
x = 4/3
(4/3, 0)
• 2x + 5y + 1 = 0
2x = -1 – 5y
y=0
x = -1/2
(-1/2, 0)
• x=¾y+5
y=0
x=5
(5, 0)
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Find the y intercept
• 3x – y = 4
• 2x + 5y + 1 = 0
• x=¾y+5
39
Solution
• 3x – y = 4
x=0
y = -4
(0, -4)
• 2x + 5y + 1 = 0
x=0
5y + 1 = 0, y = -1/5
(0, -1/5)
• x=¾y+5
x=0
¾ y = -5, y = -20/3
(0, -20/3)
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Using the Intercept to Graph
• 2x + 5y + 1 = 0
• Intercepts are: (-1/2, 0) and (0, -1/5)
• These can be two of your three points
41
Graph Using Intercepts
• y=x–3
Intercepts are (0, -3) and (3, 0)
• Use one more point as a check:
y = 1, x = 4
42
Graph Using Intercepts
• 3x – 4y = 12
43
Solution
• 3x – 4y = 12
Intercepts are (0, -3), (4, 0)
One additional point: x = 2, y = -3/2
44
Vertical Lines
• x = 5 is a vertical line through x = 5
it has no y intercept
45
Horizontal Lines
• y = 4 is a horizontal line through y = 4
it has no x intercept
46
Are the Following Lines Horizontal, Vertical, or Neither?
• 3x = 4x + 2
• y = 4-3x
• 4 + 4y = y + 3
• 2y + x = 2y + 4
• x=x
47
Solutions
• 3x = 4x + 2 Vertical
• y = 4-3x Line, neither vertical or horizontal
• 4 + 4y = y + 3 Horizontal Line
• 2y + x = 2y + 4 Vertical line (x drops out)
• x = x Not a line at all
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Chapter 10.4
Slope and Rate of Change
Background
• We can often relate the behavior of two quantities by a line
– The amount you eat to the amount you weigh
– The amount of sun to the temperature
– The gas setting on the stove to how fast the kettle boils
• Today we address the relative change of the two quantities
we are examining
An Example
90
80
60
50
40
30
90
20
80
10
70
2008 2009
Year
A
2010 2011
60
50
40
160
30
140
20
120
10
0
2006
2007 2008
2009 2010
Year
2011
Stock Price
0
2006 2007
Stock Price
Stock Price
70
100
80
60
40
B
Which Company’s Stock is
Growing Fastest?
20
0
2006 2007 2008 2009 2010 2011
Year
C
51
An Example
90
80
60
50
40
30
90
20
80
10
70
2008 2009
Year
A
2010 2011
60
50
40
160
30
140
20
120
10
0
2006
2007 2008
2009 2010
Year
2011
Stock Price
0
2006 2007
Stock Price
Stock Price
70
100
80
60
40
B
Hint: Examine the axes; They
are not the same
20
0
2006 2007 2008 2009 2010 2011
Year
C
52
How can we compare growth?
• In this case, the growth is illustrated by comparing the
steepness of the line
• When trying to compare the steepness of a line, we look at a
quantity we define as slope
Slope m = change in y / change in x
= vertical change / horizontal change
Could we do it the opposite? Yes, but we are used to thinking
of things going up and down.
53
Finding Slope
• Pick two points (x1, y1) and (x2, y2)
• Find
𝑦1−𝑦2
𝑥1−𝑥2
NOTE: You have to keep the order, 1 and 2, the same in the
numerator and the denominator!!
160
90
140
80
120
Stock Price
Stock Price
70
60
50
40
30
80
60
40
20
20
10
0
2006
100
2007 2008
2009 2010
2011
0
2006 2007 2008 2009 2010 2011
Year
Year
B
C
54
Sample Points
• In company B, point 1 is (2007, 20), point 2 is (2009, 80)
• In company C, point 1 is (2007, 20), point 2 is (2009, 140)
Company B: Slope = 60/2 = 30
Company C: Slope = 120/2 = 60 – far larger slope
160
90
140
80
120
Stock Price
Stock Price
70
60
50
40
30
80
60
40
20
20
10
0
2006
100
2007 2008
2009 2010
2011
0
2006 2007 2008 2009 2010 2011
Year
Year
B
C
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Slope
• The ratio of the change in y (height) to the change in
x (distance)
• We use the letter m to indicate slope
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Example
Find the slope of a line that passes through the following points:
(-5, 8) and (3, -2)
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Solution
Find the slope of a line that passes through the following points:
(-5, 8) and (3, -2)
8−(−2) −10 −5
=
=
−5−3
8
4
58
Does it matter which point is 1 and which is 2?
Find the slope of a line that passes through the following points:
(-5, 8) and (3, -2)
8−(−2)
−5−3
−2−8
3−(−5)
=
−10
8
=
−10
8
=
−5
4
=
−5
:
4
No difference!
59
Example
Find the slope:
Point 1: (-2, 0)
Point 2: (0,5)
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Solution
Point 1: (-2, 0)
Point 2: (0,5)
m = 5/2
Does it matter which points we choose?
• In the previous chart, we had coordinates
(0, 5), (-2,0) and (-4, -5)
• Let’s try to find the slope:
From the first two:
5−0
=
0−(−2)
From the last two:
0−(−5)
=
−2−(−4)
From the first and last:
5/2
5/2
5−(−5)
=
0−(−4)
10/4 = 5/2
• Any pair of points on a line give the same slope.
What does the sign of a slope mean?
• Positive slope:
• Negative slope:
63
Sign of a slope
• Positive slope:
The line is going uphill
• Negative slope:
The line is going downhill
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Slope from a Table
x
0
5
10
15
20
25
30
y
-4
-2
0
2
4
6
8
m =?
65
Can use any two points
m =?
x
0
5
10
15
20
25
30
y
-4
-2
0
2
4
6
8
m = 2/5
66
Standard Form
We can always put a linear equation into the form:
y = mx + b, where m and b are numbers
It turns out that if the equation is in this form, m is the slope!
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Example of Slope
• y = 4x – 8
• Find two points on the line:
if y = 0, then x = 2; (2, 0)
if x = 0, the y = -8; (0, -8)
• Find slope: (0 – (-8))/ (2 – 0) = 8/2 = 4
• In the form y = mx + b = 4x =8, 4 is the slope
More General Derivation
y= mx + b
Let two points on the line be (x1, y1) and (x2, y2)
y1 = m(x1) + b
y2 = m(x2) + b
y1 – y2 = m (x1 – x2) + (b – b)
(y1 – y2) = m (x1 – x2)
(y1 – y2) / (x1 – x2) = m, the slope
Example
• -2x + 3y = 11
• Write in the form y = mx + b
• -2x + 3y = 11
add 2x to each side: 3y = 2x + 11
divide both sides by 3: y = 2/3 x + 11/3
• Y = 2/3 x + 11/3, m = 2/3, the slope
Find the slope
• y = 5x -4
• 3y = -x + 5
• 2y = 4x + 8
71
Solution
• y = 5x -4; m = 5
• 3y = -x + 5; m = -1/3
• 2y = 4x + 8 m = 2
72
Finding a Slope from a Line
• Need to find the coordinates of two points on a line
4
3
2
1
y
0
-2
-1
-1 0
1
2
-2
-3
-4
x
• (-1, -3) (0,0)(1, 3)
73
Finding Slope
• (-1, -3) (0,0)(1, 3)
• (y1 – y1) / (x1 – x2) = m
• (0-(-3)) / (0-(-1)) = 3/1 = 3
Slope is 3. Can check with other points
Horizontal Line
Slope is Zero
Choose any point: the y value is the same
y1 – y2 = 0; 0/n = 0
Y = constant
75
Horizontal Line
• Slope zero means m = 0
• e.g., y = 5
76
Vertical Line
• Slope is
𝑦1−𝑦2
𝑥1−𝑥2
All values of x are the same; x1 – x2 = 0
The denominator is 0, so the fraction, and the slope are
undefined
77
Vertical Line
• x=5
• If we have a constant x, x cannot change.
– Means x1 – x2 = 0 for all values of x
–
𝑦1−𝑦2
,
𝑥1−𝑥2
x1-x2=0, so we have
𝑛
0
– Slope is undefined!
78
Recap
• The slope of a line tells how steep it is
– Larger slope is steeper
– Negative slope is downhill
• To find the slope, take two points on the line and calculate the
difference in the y values, and divide by the difference in the
x values:
𝑦1−𝑦2
𝑥1−𝑥2
= m, the slope
– Be sure you keep the points in the right order!
• Standard form is
y = mx + b, where m is the slope
79
More Recap
• A horizontal line, y= const, has slope zero
– It doesn’t go up or down
• A vertical line, x = const, has an infinite, or undefined slope
– It goes up faster than we can calculate
80
Review Examples
Find the slope:
• 3x – 5y = 1
• x = -4y
• -4x – 7y = 9
• x=1
• y = -3
Solutions
• 3x – 5y = 1; m = 3/5
• x = -4y; m = -1/4
• -4x – 7y = 9; m = - 4/7
• x = 1; mis undefined
• y = -3; m=0
Find the slope of the following lines
• y = 5x – 3
• 3y = 4x
• 2y – x = 4 – x
• 3x = y -4
• 5y = x + 5y + 4
83
Solutions
• y = 5x – 3
m=5
• 3y = 4x
m = 4/3
• 2y – x = 4 – x
2 y = 4, horizontal line, slope zero: m = 0
• 3x = y -4
m=3
• 5y = x + 5y + 4
0 = x + 4, x = -4, vertical line, undefined slope, m undefined
84
Deriving a Linear Equation
85
Linear Equations
• Standard form: y = mx + b
m is the slope
b is the y intercept, where the line intersects the x axis
x=0
• Can find the equation for a line given
A. A Slope and a Point
B. Two points (from two points can find a slope, then do A)
86
Example
• Find the line with slope 2 that contains the point (4, 6)
87
Solution
• Find the line with slope 2 that contains the point (4, 6)
• y = mx + b
• Substitute for m: y = 2x + b
• Put in 4 and 6: 6 = 2(4) + b = 8 + b
b = -2 since 6 = 8 – 2
• y = 2x - 2
88
Example
• Find the line with slope = -1/2 that passes through the
point (0, 4)
89
Solution
• Find the line with slope = -1/2 that passes through the
point (0, 4)
• y = mx + b; substitute for m
y = -x/2 + b
• Put in the point (0, 4)
4 = -(0)/2 + b = 0 + b
b=4
• y = -x/2 + 4
• Can check your answer:
4 = -0/2 + 4
90
Example
• Find the equation of line that contains the points
(3, 5) and (6, -1)
91
Solution
• Find the equation of line that contains the points
(3, 5) and (6, -1)
• m = (y1 – y2) / (x1 – x2)
= (5 – (-1))/(3 – 6)= 6/(-3) = -2
• y = -2x + b
put in either of the points given
• 5 = -2(3) + b = -6 + b
b = 11
• y = -2x + 11
• Check solution: -1 = -2(6) + 11 = -12 + 11 = -1
5 = -2(3) + 11 = -6 + 11 = 5
92
Example
• Find the equation of the line that contains the points
(4, -5) and (-3, 1)
93
Solution
• Find the equation of the line that contains the points
(4, -5) and (-3, 1)
• m = (-5 – 1)/(4 – (-3)) = -6/7
• y = -6/7 x + b
• Plug in points:
1 = (-6/7)(-3) + b = -18/7 + b
1 - 18/7 = 7/7 - 18/7 = -11/7 = b
• y = -6/7x - 11/7
• Check: -5 = (-6/7)(4) - 11/7 = -24/7 - 11/7 = -35/7= -5
1 = (-6/7)(-3) – 11/7 = 18/7 – 11/7 = 7/7 = 1
94
Examples
• (0, 4) and (5, -1) are on a line: find the equation of the line
95
Solution
• (0, 4) and (5, -1) are on a line: find the equation of the line
• Find slope:
96
Some More Special Lines
97
Parallel Lines
• Parallel lines have the same slope, yet different intercepts
• 5y = 3x – 4
• 5y = 3x – 8
• Let’s graph these lines
98
2
1
y
0
-5 -4 -3 -2 -1
-1 0 1 2 3 4 5
-2
-3
-4
-5
x
5y = 3x - 4
5y = 3x - 8
Perpendicular Lines
• Perpendicular lines have slopes that are negative reciprocals
• Line 1 has slope 3, Line 2 has slope – 1/3. These are
perpendicular lines
• 3x = y + 5
• x = -3y + 10
100
Perpendicular Example
• 3x = y + 5
m=3
• x = -3y + 10
m = - 1/3
(-1/3 ) ( 3) = 1
101
Graph these lines using slope
• 3x = y + 5
• x = -3y + 10
Solution
• 3x = y + 5; same as y = 3x - 5
• x = -3y + 10; same as y = -x/3 + 10/3
8
4
x = -3y + 10
3x = y + 5
y
0
-8
-4
0
-4
-8
x
4
8
Are these equations parallel, perpendicular, or neither
• y = 4x – 2
4x + y = 5
• 6 + 4x = 3y
3x + 4y = 8
• 6x = 5y + 1
-12x + 10y = 1
Solution
• y = 4x – 2
4x + y – 5 neither
• 6 + 4x = 3y
3x + 4y = 8 perpendicular
• 6x = 5y + 1
-12x + 10y = 1 parallel
Review: Solve each equation for y
• y – 7 = -9 ( x – 6 )
• y – ( - 3) = 4 (x – (- 5))
Section 10.5
Equations of Lines
Using the Intercept to Graph
• 2x + 5y + 1 = 0
• Intercepts are: (-1/2, 0) and (0, -1/5)
• These can be two of your three points
108
Graph Using Intercepts
• y=x–3
Intercepts are (0, -3) and (3, 0)
• Use one more point as a check:
y = 1, x = 4
109
Graph Using Intercepts
• 3x – 4y = 12
110
Solution
• 3x – 4y = 12
Intercepts are (0, -3), (4, 0)
One additional point: x = 2, y = -3/2
111
Graphing Given a Point and a Slope
• Find the point
• Count up and over, depending on the slope
• For example: graph the line with slope 3 that contains (1,2)
1. Plot the point (1,2)
2. Count up 3 over 1 from that point to (2, 5)
NOTE: If the slope is a whole number, the denominator is 1
Example
• Graph the line that passes through (-2, 3) with slope 5
113
Solution
Graph the line that passes through (-2, 3) with slope 5
1. Plot the point (-2, 3)
2. Count up 5 and over 1 to (-1, 8) and connect the points
114
Example
• Graph a line with slope 1/3 that passes through (-1, -2)
115
Solution
• Graph a line with slope 1/3 that passes through (-1, -2)
1. Plot the point (-1, -2)
2. From there go up 3 and over 1 to get to (1, 1)
3. Connect the points
116
Example
• Graph the line of slope – 4/3 that passes through (-1, 8)
117
Solution
• Graph the line of slope – 4/3 that passes through (-1, 8)
1. Plot the point (-1, 8)
2. From that point go down 4 and over 3 to get to (2, 4)
3. Connect the points
• Note: we could have also gone up 4 and to the left three from
our point and reached (-5, 12). This point is also on the line
118
Graph from a Slope and an Intercept
• 4x + y = 1
• First put in format y = mx + b
y = -4x + 1
• y Intercept is (0,1)
• Slope, m is -4
• Find intercept, and count down 4 and over 1
Example
• 4x + y = 1
• y= - 4x+1
intercept is (0, 1), slope is – 4
120
What you need to know how to do:
• Given the points, find the slope and write the equation
• Given the slope and a point, write the equation
• Given the slope and the y intercept, write the equation
• And, how to graph these equations