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1.2: Slope
-slope formula
M(G&M)–10–9 Solves problems on and off the coordinate plane involving distance, midpoint, perpendicular and parallel lines, or slope.
GSE
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Understanding Slope
• If a line rises as you move from left to right, then the slope is positive.
2
-2
F: (1, 2)
E: (-2, -2)
Riding a bike uphill
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Understanding Slope
• If a line drops as you move from left to right, then the slope is negative.
4
2
H: (1, 2)G: (-2, 3)
Skiing Downhill
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Understanding Slope
• A horizontal line has zero slope: m = 0 2
K: (2, 1)J: (-2, 1)
Running on a flat surface like a track Or any athletic fieldRunning on a flat surface like a track Or any athletic fieldRunning on a flat surface like a track Or any athletic field
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Understanding Slope
• A vertical line has no slope: m is undefined.
4
2
N: (2, 1)
M: (2, 3)
Running into a wall, youcant get past itRunning into a wall, youcant get past it
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Slope Formula
The slope of a line through the points (x1, y1) and (x2, y2) is as follows:
yy22 –– yy11 xx22 –– xx11
m =
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Ex.
Find the slope of the line that passes through (–2, –3) and (4, 6).
Let (x1, y1) be (–2, –3) and (x2, y2) be (4, 6).
6 – (–3)4 – (–2)
Substitute 6 for y2, –3 for y1, 4 for x2, and –2 for x1.
=y2 – y1
x2 – x1
96= 3
2=
The slope of the line that passes through (–
2, –3) and (4, 6) is . 32
*** Always reduce your fractions****
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Understanding Slope
• Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.)
4
2
-2
-4
-6
5
D: (4, -1)
C: (-2, -4)
B: (3, 3)
A: (-1, 1)
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Understanding Slope
• The slope of AB is:
• The slope of CD is:
• Since m1=m2, AB || CD
4
2
-2
-4
-6
5
D: (4, -1)
C: (-2, -4)
B: (3, 3)
A: (-1, 1) 1
3 1 2 1
3 1 4 2m
2
1 4 3 1
4 2 6 2m
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Perpendicular Lines
• (┴)Perpendicular Lines- 2 lines that intersect forming 4 right angles
Right angle
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Slopes of Lines
• In a coordinate plane, 2 non vertical lines are iff the product of their slopes is -1.
• This means, if 2 lines are their slopes are opposite reciprocals of each other; such as ½ and -2.
• Vertical and horizontal lines are to each other.
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Example• Line l passes through (0,3) and (3,1).
• Line m passes through (0,3) and (-4,-3).
Are they ?
Slope of line l =
Slope of line m =
l m
30
13
3
2-or
3
2
40
33
2
3or
4
6
Opposite Opposite Reciprocals!Reciprocals!
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Equation of a line in slope intercept form (y = mx+b)
Now that we know how to find slope given any two points, we cangenerate an equation of the line connecting the two points.
Example : points (3,2) and (6,9)
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2nd example
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Slope-Intercept Form (y = mx+b)
• Find the equation of a line passing through the points P(0, 2) and Q(3, –2).
2
-2
Q: (3, -2)
P: (0, 2)
•Is this line parallel to a line with the equation
43?
3y x
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a) Find the equation of a line that passes through the points G ( -4, 5) and H (-8, 3)
b) Write the equation of a line that passes through point P (1, -2) and is perpendicular to the one from part a
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Slope Graphically
You can always count ! (not suggested as you advance In your math courses)
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Homework
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Another application of Slope
run
rise
Slope is rise
run
The steepness of the ramp matters to people who need to walk on it.
or Rise:Run
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No minimum pitch in the code.
Required slope would be determined by roofing materials.
Most shingle type roofs require minimum 4 in 12 pitch.
You can go as low as 2.5 in 12 with special underlayment.
Some local jurisdictions with heavy snowfall require a steep pitched roof for obvious reasons.
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Rhode Island Code
• The Maximum slope for wheel chair ramps is 1:8 .
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The house has a platform 6 ft off the ground. If RI code says the maximum slope is 1:8What could the lengths of the run be for a ramp?
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Assignments