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2.3 – Slope
2.3 – Slope
Slope
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s
change in x’s
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s
run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s
run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s
run
m = y2 – y1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
2.3 – Slope
Slope – ratio of the difference in the
y-coordinates to the difference in the
x-coordinates
slope = change in y’s = rise
change in x’s run
m = y2 – y 1
x2 - x1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 –
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 –
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 – x1
= -2 – 4
1–
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1–
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2 1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2 1
3
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2 1
3
1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2 1
3
1
3
1
Example 1 Find the slope of the line passing
through (-1,4) and (1,-2) and graph.
m = y2 – y1
x2 - x1
= -2 – 4
1 – (-1)
= -6 = -3
2 1
3
1
3
1
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
2
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
3
2
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
3
2
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
3
2
Example 2 Graph the line passing through
(-4,-3) with a slope of ⅔.
3
2
3
2
• Parallel Lines
• Parallel Lines – have the same slope!
• Parallel Lines – have the same slope!
• Perpendicular Lines
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical
• Parallel Lines – have the same slope!
slope = m
• Perpendicular Lines – the slope is the opposite reciprical
• Parallel Lines – have the same slope!
slope = m
• Perpendicular Lines – the slope is the opposite reciprical.
slope = -1
m
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
• Parallel Lines – have the same slope!
• Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to
the line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
m1 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
1
y = -x – 4
4 4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
1
y = -x – 4
4 4
4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
1
y = -x – 4
4 4
4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
•
•
Parallel Lines – have the same slope!
Perpendicular Lines – the slope is the opposite reciprical, i.e. -1
m
Example 3 Graph the line that passes through (-1,3) that is parallel to the
line with equation x + 4y = -4.
Slope-Intercept form: y = mx + b, where m = slope, b = y-int.
(y= form)
x + 4y = -4
-x
-x
4y = -x – 4
4
4
1
y = -x – 4
4 4
4
y = -¼x – 1
y = mx + b
m1 = -¼
m2 = -¼
Example 4 Graph the line through (-3,1)
that is perpendicular to the line with
equation 2x + 5y = -10.
Example 4 Graph the line through (-3,1)
that is perpendicular to the line with
equation 2x + 5y = -10.
*Find slope of line, then find opp. reciprical!
Example 4 Graph the line through (-3,1)
that is perpendicular to the line with
equation 2x + 5y = -10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
m1 = -2/5
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2
Example 4 Graph the line through (-3,1) that is
perpendicular to the line with equation 2x + 5y =
-10.
*Find slope of line, then find opp. reciprical!
2
2x + 5y = -10
-2x
-2x
5y = -2x – 10
5
5
5
5
y = -2 x – 2
5
m1 = -2/5
m2 = 5/2