3.6 and 3.7 Lines 2010
October 21, 2010
3.6 Lines in the Coordinate Plane
&
3.7 Slopes of Parallel and Perpendicular Lines
Objectives:
Graph lines given their equations.
Write equations of lines.
Relate slope and parallel lines.
Relate slope and perpendicular lines.
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3.6 and 3.7 Lines 2010
October 21, 2010
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
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3.6 and 3.7 Lines 2010
October 21, 2010
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
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y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
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3.6 and 3.7 Lines 2010
October 21, 2010
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
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3.6 and 3.7 Lines 2010
October 21, 2010
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3.6 and 3.7 Lines 2010
October 21, 2010
Review: Here is a formula for finding the slope of a line.
Slope Formula:
m =
y2 ­ y1
x2 ­ x1
Example: Find the slope of the line through the points (3,2) and (­9,6).
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3.6 and 3.7 Lines 2010
October 21, 2010
Find the slope of the line that contains each pair of points.
2. X(3, 0) and Y(0, ­5)
1. A(­2, 2) and B(4, ­3)
3. R(0, 1) and S(­3, 5)
4. L(7, ­10) and M(1, ­4)
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3.6 and 3.7 Lines 2010
October 21, 2010
Linear Function
• Graph is a line.
• Example: y = 3x + 2
• Solution: any ordered pair (x, y) that makes the equation true. (any point on the line)
• The value of y depends on the value of x.
• Therefore, y is the dependent variable and x is the independent variable.
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3.6 and 3.7 Lines 2010
October 21, 2010
The y­intercept of a line is the point at which the line crosses the y­axis.
The x­intercept of a line is the point at which the line crosses the x­axis.
(0, y) y­intercept
(x, 0) x­intercept
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3.6 and 3.7 Lines 2010
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Graphing a Linear Equation
There are many ways to graph a line. Here is one method.
Example:
2
Graph the equation y = x + 3
3
Graph by plotting points.
x
y
­3
0
3
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Slope­Intercept Form:
y = mx + b
y = mx + b
slope y­intercept
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3.6 and 3.7 Lines 2010
October 21, 2010
Graphing a Linear Equation
There are many ways to graph a line. Here is another method.
Example:
1
Graph the equation: y = x ­ 5
4
Graph by using slope­intercept form.
Slope­Intercept Form:
Slope:
y­intercept:
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3.6 and 3.7 Lines 2010
October 21, 2010
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3.6 and 3.7 Lines 2010
October 21, 2010
Example:
2
Graph the equation: y = ­ x ­ 3
5
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3.6 and 3.7 Lines 2010
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Standard Form:
Ax + By = C
Ax + By = C
Positive Integer Integer
Integer
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3.6 and 3.7 Lines 2010
October 21, 2010
Graphing a Linear Equation
There are many ways to graph a line. Here is another method.
Example:
Graph the equation: 3x ­ 2y = 18.
Graph by finding intercepts.
x
y
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3.6 and 3.7 Lines 2010
October 21, 2010
Example: Write in slope­intercept form to find the slope of 4x + 3y = 12 and graph the line.
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3.6 and 3.7 Lines 2010
October 21, 2010
Example: Write in slope­intercept form to find the slope of ­5x + y = ­3 and graph the line.
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3.6 and 3.7 Lines 2010
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Point-Slope Form:
The line through point (x1, y1) with
slope m has the equation:
y - y1 = m(x - x1)
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3.6 and 3.7 Lines 2010
October 21, 2010
Example: Write in point­slope form an 1
equation of the line with slope ­ 2 through the point (8, ­1).
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3.6 and 3.7 Lines 2010
October 21, 2010
Example: Write in point­slope form the equation of the line through (1, 5) and (4, ­1).
y ­ 5 = ­2(x ­ 1)
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3.6 and 3.7 Lines 2010
October 21, 2010
Review: Equations of a Line
Match each name with the correct equation and appropriate example.
Point-Slope Form
y - y1 = m(x - x1)
y - 2 = -3(x + 4)
Standard Form
Ax + By = C
3x + y = -10
Slope-Intercept Form
y = mx + b
y = -3x - 10
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3.6 and 3.7 Lines 2010
October 21, 2010
Attention:
The slopes of
horizontal and
vertical lines
have special
properties.
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October 21, 2010
Horizontal Line:
y=b
m=0
y is constant
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3.6 and 3.7 Lines 2010
October 21, 2010
Vertical Line:
x=c
m is undefined
x is constant
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3.6 and 3.7 Lines 2010
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Example: Write the equations of the horizontal and vertical lines that contain the point P(5, ­1).
y
6
5
4
3
2
1
­6
­5
­4
­3
­2
­1
0
­1
x
1
2
3
4
5
6
­2
­3
­4
­5
­6
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Parallel Lines:
y = mx + b1
y = mx + b2
m1 = m 2
b 1 ≠ b2
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Example:
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3.6 and 3.7 Lines 2010
October 21, 2010
Example:
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3.6 and 3.7 Lines 2010
October 21, 2010
Example:
Write an equation of the line through the
point (-2, 4) and parallel to y = (- 12 )x + 2.
Then graph both lines.
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3.6 and 3.7 Lines 2010
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Perpendicular Lines:
y = m1x + b1
y = m2x + b2
m1(m2) = -1
m1 and m2 are opposite reciprocals
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3.6 and 3.7 Lines 2010
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Example:
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3.6 and 3.7 Lines 2010
October 21, 2010
Example:
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3.6 and 3.7 Lines 2010
October 21, 2010
Example:
Write an equation of the line through the
point (6, 1) and perpendicular to y = 34 x + 2.
Then graph both lines.
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3.6 and 3.7 Lines 2010
October 21, 2010
Homework:
page 169 (1 ­ 4, 11 ­ 37 odd) &
page 177 (9 ­ 15, 20 ­ 22, 25 ­ 27)
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