36 slopes of lines-x
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Slopes of Lines
Slopes of LinesThe steepness of a street is measured in “grade”.
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
The 18%-grade means the ratio of 18 to100 as shown here:
18 ft100 ft
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
The 18%-grade means the ratio of 18 to100 as shown here:
18 ft100 ft
The steepness of a roof is measured in “pitch”.
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
The 18%-grade means the ratio of 18 to100 as shown here:
18 ft100 ft
The steepness of a roof is measured in “pitch”.For example:
(12”)
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
The 18%-grade means the ratio of 18 to100 as shown here:
18 ft100 ft
Here is outline of a roof with a pitch of 4:12 or 1/3.
(12”)
The steepness of a roof is measured in “pitch”.For example:
Slopes of Lines
a Seattle trolleybus climbing an 18%-grade street (Wikipedia)
The steepness of a street is measured in “grade”. For example:
The 18%-grade means the ratio of 18 to100 as shown here:
18 ft100 ft
Here is outline of a roof with a pitch of 4:12 or 1/3.
(12”)
The steepness of a roof is measured in “pitch”.For example:
In mathematics, these measurements are called “slopes”.
Slopes of LinesThe slope of a line is a number.
Slopes of LinesThe slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.
Slopes of LinesThe slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of LinesThe slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Definition of Slope
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics.
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5,
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5, then Δy = y2 – y1
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4,
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4, then Δ x = x2 – x1
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Slopes of Lines
Definition of SlopeNotation: The Greek capital letter Δ (delta) in generalmeans “the difference” in mathematics. Δy means the difference in the values of y’s, Δx means the difference the values of x’s.Example A. Let y1 = –2, y2 = 5, then Δy = y2 – y1 = 5 – (–2) = 7
Let x1 = 7, x2 = –4, then Δ x = x2 – x1 = –4 – 7 = –11
The slope of a line is a number. The slope of a line measures the amount of tilt, (inclination, steepness) of the line against the x-axis.Steep lines have slopes with large absolute value. Gradual lines have slopes with small absolute value
Definition of Slope Slopes of Lines
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line,
Slopes of Lines
(x1, y1)
(x2, y2)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔxm =
Slopes of Lines
(x1, y1)
(x2, y2)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
Geometry of Slope
(x1, y1)
(x2, y2)
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”.
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δx
easy to memorize
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1x2 – x1
=
easy to memorize
the exact formula
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1x2 – x1
=
easy to memorize
the exact formula
the geometric meaning
Definition of Slope Let (x1, y1) and (x2, y2) be two points on a line, then the slope m of the line is
ΔyΔx
y2 – y1
x2 – x1m = =
riserun=
Slopes of Lines
(x1, y1)
(x2, y2)
Δy=y2–y1=rise
Δx=x2–x1=run
Geometry of Slope Δy = y2 – y1 = the difference in the heights of the points.Δx = x2 – x1 = the difference in the runs of the points.
ΔyΔx= Therefore m is the ratio of the “rise” to the
“run”. m = Δy
Δxy2 – y1x2 – x1
=
easy to memorize
the exact formula
the geometric meaning
Note that y2 – y1
x2 – x1
y1 – y2
x1 – x2=
so the numbering of thetwo points is not relevant– we get the same answer.
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
(–2 , 8)( 3 , –2)
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
(–2 , 8)( 3 , –2)–5 , 10
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
Δy
(–2 , 8)( 3 , –2)–5 , 10
Δx
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
(–2, 5)( 3, 5)
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
Δy
(–2, 5)( 3, 5)–5, 0
Δx
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
Δy
(–2, 5)( 3, 5)–5, 0
Δx
So the slope is
Δx Δym =
Δy
Δx
(–2 , 8)( 3 , –2)–5 , 10
Δy
Δx
Hence the slope is10–5
Example B. Find the slope of the line that passes through (3, –2) and (–2, 8). Draw the line.
Slopes of Lines
It’s easier to find Δx and Δy vertically.
m =
=
= –2
Example C. Find the slope of the line that passes through (3, 5) and (-2, 5). Draw the line.
Δy
(–2, 5)( 3, 5)–5, 0
Δx
So the slope is
Δx Δy 0
–5 m =
=
= 0
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
So the slope
Δx Δy 3
0 m =
=
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
So the slope
Δx Δy 3
0 m =
=
is undefined!
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
So the slope
Δx Δy 3
0 m =
=
is undefined!
Hence the slope mof a horizontal line is m = 0.
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
So the slope
Δx Δy 3
0 m =
=
is undefined!
Hence the slope m of a vertical line is undefined (UDF).of a horizontal line is m = 0.
As shown in example C, the slope of a horizontal line is 0, i.e. it’s “tilt” is 0.
Slopes of Lines
Example D. Find the slope of the line that passes through (3, 2) and (3, 5). Draw the line.
Δy
(3, 5)(3, 2)0, 3
Δx
So the slope
Δx Δy 3
0 m =
=
is undefined!
Hence the slope m of a vertical line is undefined (UDF).of a horizontal line is m = 0.
of a tilted line is a non–zero number.
Exercise A. Identify the vertical and the horizontal lines by inspection first. Find their slopes or if it’s undefined, state so. Fine the slopes of the other ones by solving for the y.1. x – y = 3 2. 2x = 6 3. –y – 7= 0
4. 0 = 8 – 2x 5. y = –x + 4 6. 2x/3 – 3 = 6/5
7. 2x = 6 – 2y 8. 4y/5 – 12 = 3x/4 9. 2x + 3y = 3
10. –6 = 3x – 2y 11. 3x + 2 = 4y + 3x 12. 5x/4 + 2y/3 = 2 Exercise B. 13–18. Select two points and estimate the slope of each line.
13. 14. 15.
More on Slopes
16. 17. 18.Exercise C. Draw and find the slope of the line that passes through the given two points. Identify the vertical line and the horizontal lines by inspection first.19. (0, –1), (–2, 1) 20. (1, –2), (–2, 0) 21. (1, –2), (–2, –1)22. (3, –1), (3, 1) 23. (1, –2), (–2, 3) 24. (2, –1), (3, –1)25. (4, –2), (–3, 1) 26. (4, –2), (4, 0) 27. (7, –2), (–2, –6)28. (3/2, –1), (3/2, 1) 29. (3/2, –1), (1, –3/2)30. (–5/2, –1/2), (1/2, 1) 31. (3/2, 1/3), (1/3, 1/3)32. (–2/3, –1/4), (1/2, 2/3) 33. (3/4, –1/3), (1/3, 3/2)
More on Slopes
Exercise D. 34. Identify which lines are parallel and which one are perpendicular. A. The line that passes through (0, 1), (1, –2)
D. 2x – 4y = 1
B. C.
E. The line that’s perpendicular to 3y = xF. The line with the x–intercept at 3 and y intercept at 6. Find the slope, if possible of each of the following lines.35. The line passes with the x intercept at x = 2, and y–intercept at y = –5.
More on Slopes
36. The equation of the line is 3x = –5y+737. The equation of the line is 0 = –5y+7 38. The equation of the line is 3x = 739. The line is parallel to 2y = 5 – 6x 40. the line is perpendicular to 2y = 5 – 6x41. The line is parallel to the line in problem 30. 42. the line is perpendicular to line in problem 31.43. The line is parallel to the line in problem 33. 44. the line is perpendicular to line in problem 34.
More on SlopesFind the slope, if possible of each of the following lines
Summary of SlopeThe slope of the line that passes through (x1, y1) and (x2, y2) is
Horizontal line Slope = 0
Vertical line Slope is UDF.
Tilted line Slope = –2 0
riserun= m = Δy
Δxy2 – y1
x2 – x1=
Exercise A. Select two points and estimate the slope of each line.
1. 2. 3. 4.
Slopes of Lines
5. 6. 7. 8.
Exercise B. Draw and find the slope of the line that passes through the given two points. Identify the vertical line and the horizontal lines by inspection first.9. (0, –1), (–2, 1) 10. (1, –2), (–2, 0) 11. (1, –2), (–2, –1)12. (3, –1), (3, 1) 13. (1, –2), (–2, 3) 14. (2, –1), (3, –1)15. (4, –2), (–3, 1) 16. (4, –2), (4, 0) 17. (7, –2), (–2, –6)18. (3/2, –1), (3/2, 1) 19. (3/2, –1), (1, –3/2)20. (–5/2, –1/2), (1/2, 1) 21. (3/2, 1/3), (1/3, 1/3)22. (–2/3, –1/4), (1/2, 2/3) 23. (3/4, –1/3), (1/3, 3/2)
Slopes of Lines
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