4.5 Finding The Slope of a Line
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4.5 Finding The Slope of a
Line
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What is the meaning of this sign?
1. Icy Road Ahead
2. Steep Road Ahead
3. Curvy Road Ahead
4. Trucks Entering Highway Ahead
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Slopes are commonly associated with mountains.
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The slope we are studying is associated with the graph of a
line.
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Definitions of Slope
• The tilt or inclination of a line
• The ratio of vertical change to horizontal change.
• The change in y over the change in x
ΔyΔx
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Tilt or Inclination
FROM LEFT TO RIGHT!!!!
POSITIVE SLOPE
NEGATIVESLOPE
SLOPE IS ZERO
SLOPE ISUNDEFINED
y = #
x = #
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Tell me slope of each line
POSITIVE NEGATIVE
ZEROPOSITIVE
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Slope can be expressed different ways:
A line has a positive slope if it is going uphill from left to right.
A line has a negative slope if it isgoing downhill from left to right.
2 1
2 1
( ) vertical change
( ) horizontal change
y y risem
x x run
UP or DOWN RIGHT
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Graph (3,2) and (-1,-1)
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Draw a line through the points.
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Rise
3
4
Run
THE SLOPE IS
34
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Find the slope of the following line.
The slope is…
1
2
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Find the slope of the line.
The slope is….
-3
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When given the graph, it is easier to apply “rise over run”.
1) Determine the slope of the line.
€
3
6=
1
2
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Determine the slope of the line shown.
1. -2
2. -½
3. ½
4. 2
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Determine the slope of the line.
The line is decreasing (slope is negative).
-2
1
rise
run
2
1 2
Find points on the graph. Use two of them and apply rise over run.
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What is the slope of a horizontal line?
The line doesn’t rise!
m 0
number0
All horizontal lines have a slope of 0.
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What is the slope of a vertical line?
The line doesn’t run!
All vertical lines have an undefined slope.
m number
0undefined
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Find the slope of these lines.
• Black line 3
• Red Line 1
• Blue Line -1/2
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Find the slope of these lines
• Orange line = 0
• Green Line = Undefined
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Determining Slope
Rise=1
Slope= 12
Slope=3
Rise=6Run =2
Run =2Rise=12Run =4
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Let’s find the slope of a line with two points, WITHOUT LOOKING AT A GRAPH.
REMEMBER THE SLOPE FORMULA???!!!!!
€
m =y2 − y1
x2 − x1
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Determining Slope without graphing
(-4, -6)
(2, 5)
•Pick 2 points on the line
•Find the change in the Y-coordinates by subtracting(rise)
•Find the change in the X-coordinates by subtracting(run)
5-(-6)2-(-4)
= 11 6
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2) Find the slope of the line that passes through the points (-2, -2) and (4, 1).
(1 ( 2))
(4 ( 2))m
2 1
2 1
( )
( )
y ym
x x
(1 2)
(4 2)
When given points, it is easier to use the formula!
y2 is the y coordinate of the 2nd ordered pair (y2 = 1)
y1 is the y coordinate of the 1st ordered pair (y1 = -2)
13
6 2
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3) Find the slope of the line that goes through the points (-5, 3) and (2, 1).
m y2 y1
x2 x1
1 3
2 ( 5)m
1 3
2 5m
2
7m
x1 y1 x2 y2
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Sample Problems:• Find the slope of the line thru the points given:
(-3,-1) and (-2,4)
(-3,4) and (2,-2)
m 4 1 2 3
5
15
m 2 4
2 3 6
5
x1 y1 x2 y2
x1 y1 x2 y2
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Independent Practice: Calculate the slope for each pair of points. Classify each slope as positive, negative, zero, or undefined.
1. (2, 2) and (3, 5) 2. (0, 0) and (3, 0)
3. (-2, -1) and (-1, -4)
4. (2, 3) and (2, 7)
5. (-1, -1) and (5, 5)
6. (8, 4) and (6, 4)
Slope: 3 Classification: Positive
Slope: -3
Classification: Negative
Slope: 1 Classification: Positive
Slope: 0 Classification: Zero
Slope: Undefined Classification: Vertical
Slope: 0
Classification: Horizontal
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EVERYONE GET A COMMUNICATOR!!!
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1
Draw a line through the point (2,0) that has a slope of 3.
1. Graph the ordered pair (2, 0).
2. From (2, 0), apply rise over run (write 3 as a fraction).
3. Plot a point at this location.
4. Draw a straight line through the points.
3
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If you know the slope and one point on the line you can draw the line.
Example: slope is 1/3 and the point (-2,-1)
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Now you draw a line.
• Given: slope is 2/3 and the point is (1,0)
• Given: slope is –3 and the point is (–2,4)
• Given: slope is 1 and the point is (2,3)
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Graph the line y = 3x +1,then tell me the slope of that line
after you graphed it.The slope is 3
x y
-2
-1
0
1
2
-5
-2
1
4
7
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Graph the line 2x + y = 4,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
8
6
4
2
0
The slope is -2y = -2x + 4
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Graph the line y = -5,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
-5
-5
-5
-5
-5
The slope is 0
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Graph the line y = 2x,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
-4
-2
0
2
4
The slope is 2
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Graph the line y = 3x - 2,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
-8
-5
-2
1
4
The slope is 3
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Graph the line 2x + 3y = 6,then tell me the slope of that line
after you graphed it.
x y
-6
-3
0
3
6
6
4
2
0
-2
The slope is -2/3
€
y =−2
3x + 2
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Graph the line y = -x - 2,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
0
-1
-2
-3
-4
The slope is -1
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Graph the line 3x – 4y = 12,then tell me the slope of that line
after you graphed it.
x y
-8
-4
0
4
8
-9
-6
-3
0
3
The slope is 3/4
€
€
y =3
4x − 3
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Graph the line x – y = 5,then tell me the slope of that line
after you graphed it.
x y
-2
-1
0
1
2
-7
-6
-5
-4
-3
The slope is 1y = x - 5
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The slope of a line that goes through the points (r, 6) and (4, 2) is 4. Find r.
To solve this, plug the given information into the formula
m
( y2 y
1)
(x2 x
1)
.
2 64
4 r
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4 4
1 4 r
To solve for r, simplify and write as a proportion.
4 4
1 4 r
2 6
44 r
Cross multiply.
1(-4) = 4(4 – r)
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Simplify and solve the equation.1(-4) = 4(4 – r)
-4 = 16 – 4r-16 -16
-20 = -4r-4 -45 = r
The ordered pairs are (5, 6) and (4, 2)