copyright © 2013, 2009, 2005 pearson education, inc. section 2.4 equations of lines and linear...
TRANSCRIPT
Copyright copy 2013 2009 2005 Pearson Education Inc
Section 24
Equations of Lines and Linear
Models
Copyright copy 2013 2009 2005 Pearson Education Inc
Objectives
bull Point-Slope Form
bull Horizontal and Vertical Lines
bull Parallel and Perpendicular Lines
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The line with slope m passing through the point (x1 y1) is given by
y = m(x ndash x1) + y1
or equivalently
y ndash y1 = m(x ndash x1)
the point-slope form of a line
POINT-SLOPE FORM
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the point-slope form of a line passing through the point (3 1) with slope 2 Does the point (4 3) lie on this line SolutionLet m = 2 and (x1 y1) = (31) in the point-slope form
To determine whether the point (4 3) lies on the line substitute 4 for x and 3 for y
y ndash y1 = m(x ndash x1)
y minus 1 = 2(x ndash 3)
3 ndash 1 2(4 ndash 3)
2 = 2The point (4 3) lies on the line because it satisfies the point-slope form
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Objectives
bull Point-Slope Form
bull Horizontal and Vertical Lines
bull Parallel and Perpendicular Lines
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The line with slope m passing through the point (x1 y1) is given by
y = m(x ndash x1) + y1
or equivalently
y ndash y1 = m(x ndash x1)
the point-slope form of a line
POINT-SLOPE FORM
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the point-slope form of a line passing through the point (3 1) with slope 2 Does the point (4 3) lie on this line SolutionLet m = 2 and (x1 y1) = (31) in the point-slope form
To determine whether the point (4 3) lies on the line substitute 4 for x and 3 for y
y ndash y1 = m(x ndash x1)
y minus 1 = 2(x ndash 3)
3 ndash 1 2(4 ndash 3)
2 = 2The point (4 3) lies on the line because it satisfies the point-slope form
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The line with slope m passing through the point (x1 y1) is given by
y = m(x ndash x1) + y1
or equivalently
y ndash y1 = m(x ndash x1)
the point-slope form of a line
POINT-SLOPE FORM
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the point-slope form of a line passing through the point (3 1) with slope 2 Does the point (4 3) lie on this line SolutionLet m = 2 and (x1 y1) = (31) in the point-slope form
To determine whether the point (4 3) lies on the line substitute 4 for x and 3 for y
y ndash y1 = m(x ndash x1)
y minus 1 = 2(x ndash 3)
3 ndash 1 2(4 ndash 3)
2 = 2The point (4 3) lies on the line because it satisfies the point-slope form
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the point-slope form of a line passing through the point (3 1) with slope 2 Does the point (4 3) lie on this line SolutionLet m = 2 and (x1 y1) = (31) in the point-slope form
To determine whether the point (4 3) lies on the line substitute 4 for x and 3 for y
y ndash y1 = m(x ndash x1)
y minus 1 = 2(x ndash 3)
3 ndash 1 2(4 ndash 3)
2 = 2The point (4 3) lies on the line because it satisfies the point-slope form
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Use the point-slope form to find an equation of the line passing through the points (minus2 3) and (2 5) SolutionBefore we can apply the point-slope form we must find the slope
2 1
2 1
y ym
x x
5 3
2 2
2
4
1
2
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
We can use either (minus2 3) or (2 5) for (x1 y1) in the point-slope form If we choose (minus2 3) the point-slope form becomes the following
If we choose (2 5) the point-slope form with x1 = 2 and y1 = 5 becomes
y ndash y1= m(x ndash x1)
1)3 ( )
2( 2y x
13 ( 2)
2y x
15 ( 2)
2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find an equation of the line passing through (4 2) and (1 5) SolutionFirst find the slope of the line 2 1
2 1
y ym
x x
5 2
1 4
1
3
3
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find an equation of the line passing through (4 2) and (1 5) SolutionNow substitute minus1 for m and (4 2) for x and y in the slope intercept form The point (1 5) could be used insteady mx b
1 4( )2 b
2 4 b
6 bThe slope-intercept form is y = minusx + 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find a line y = mx + b that models the data in the table
SolutionCarefully make a scatter plot of the data and sketch a line
x 1 2 3 4 5
f(x) 6 11 18 24 31
0
5
10
15
20
25
30
35
0 1 2 3 4 5 6
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find a line y = mx + b that models the data in the table
SolutionChose two points on the line and find the equation
The equation of the line is y = 6(x ndash 3) + 18 or y = 6x
x 1 2 3 4 5
f(x) 6 11 18 24 31
24 18
4 3m
6m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a The equation of a horizontal line with y-intercept b is y = b
The equation of a vertical line with x-intercept h is x = h
EQUATIONS OF HORIZONTAL AND
VERTICAL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines SolutionThe x-coordinate of the point (minus5 2) is minus5 The vertical line x = minus5 passes through every point on the xy-plane including the point (minus5 2)
Similarly the horizontal line y = 2 passes through every point with a y-coordinate of 2 including the point (minus5 2)
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find equations of the vertical and horizontal lines that pass through the point (minus5 2) Graph these two lines Solutionx = minus5y = 2
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with the same slope are parallel
Two nonvertical parallel lines have the same slope
PARALLEL LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of a line parallel to y = 3x + 1 and passing through the point (2 1)
SolutionThe line has a slope of 3 any parallel line also has slope 3
1 1( )y m x x y
3( 2) 1y x 3 6 1y x
3 5y x
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
0 and 1xf x a a a
Two lines with nonzero slopes m1 and m2 are perpendicular if m1m2 = minus1 If two lines have slopes m1 and m2 such that m1 m2 = minus1 then they are perpendicular
A vertical and horizontal line are perpendicular
PERPENDICULAR LINES
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of the line perpendicular to y = x ndash 3 passing through the point (4 6)
SolutionThe line has slope m1 = 1 The slope of the perpendicular line is m2 = minus1 The slope-intercept form of a line having slope minus1 and passing through (4 6) can be found as follows
1 1( )y m x x y
1( 4) 6y x 4 6y x
10y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y1 has the slope m1 = 32 because the line rises 3 units for every 2 units of run Itrsquos y-intercept is minus12 and itrsquos slope intercept form isThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
1
3 1
2 2y x
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m
Copyright copy 2013 2009 2005 Pearson Education Inc
Example (cont)
Find the slope-intercept form of each line shown Verify that the two lines are perpendicularSolutionThe graph of y2 has the slope m2 = minus23 because the line falls 2 units for every 3 units of run Itrsquos y-intercept is 13 and itrsquos slope intercept form is
y2
y1
2
2 1
3 3y x
1 2
3 21
2 3m m