Download Slope and Equation of a Line

Slope and Equation of a Line
1
Slope of a Line: The slope of a line is a rate of change that determines the steepness of the line. If (x1 , y1)
and (x2 , y2 ) are any two distinct points on the line, the formulas below show how to calculate a slope.
Slope of a line = m =
y - y1
y - y2
rise
change in y
Dy
=
=
= 2
= 1
run
change in x
Dx
x2 - x1
x1 - x2
Properties of Slope:
a) The slope of a horizontal line ( Dy = 0) equals zero.
b) Vertical lines ( Dx = 0) do not have a slope or we say
the slope is undefined. (Warning! No slope lines and
zero slope lines are NOT the same! )
D
C
-1
1
3
Line E: slope = 3
-2
+4
+3
-4
+1
+5
d) Lines that drop downward as you move left to right have
a negative slope.
Line C: slope = -
+6
B
Line B: slope = -1
5
Line D: slope =
2
F
+3
c) Lines that rise upward as you move left to right have
a positive slope.
Line A: slope = 0
E
+2
+6
+2
+2
-2
A
Line F: slope is undefined. (Dx = 0. Therefore Dy / Dx is undefined .)
Equation of a Line: The equation of line describes the relationship between the x-coordinate and the ycoordinate of any point on the line. Equations of lines come in five basic flavors:
a) Horizontal line: y = k where k is a constant value. The equation of line A above is y = -7.
b) Vertical line: x = k where k is a constant value. The equation of line F above is x = 8.
Line F has no slope, but it still has an equation that describes the points on the line.
c) Slope-intercept form: y = mx + b where m is the slope of the line and the point (0, b) is the y-intercept of
the line. The slope-intercept equations of lines B, C, D, and E are as follows:
Line B: y = - x – 4
Line C: y = -1/3x + 6
Line D: y = 5/2x + 6
Line E: y = 3x – 4
d) Point-slope form: y – y1 = m(x – x1 ) where m is the slope of the line, (x1 , y1 ) is any fixed point on the
line, and (x,y) represents any generic point on the line. The point-slope equations of lines B, C, and D are
as follows:
Line B: y + 5 = -(x - 1)
(Used point (1, -5) )
Line C: y – 7 = -1/3(x + 3)
Line D: y + 4 = 5/2(x + 4)
(Used point (-3, 7) )
(Used point (-4, -4) )
e) Standard form: Ax + By = C where A, B, and C are integers. The standard equation for lines B, C,
D, and E are as follows: (The equation should contain the least number of minus signs.)
Line B: x + y = -4
Line C: x + 3y = 18
Line D: -5x + 2y = 12
Line E: -3x + y = -4 or
3x – y = 4
Distance Between Two Points and Midpoint of a Line Segment
2
Distance Between Two Points: If A = (x1 , y1) and B = (x2 , y2 ) are any two distinct points, the formula
below can be used to calculate the distance from A to B.
AB =
( x1 - x2 ) 2 + ( y1 - y2 ) 2
Distance formula is a simple
application of the Pythagorean
Theorem.
A
C
Midpoint of a Segment: If A = (x1 , y1) and B = (x2 , y2 )
are the endpoints of a line segment, the formula below can be
used to find the x-y coordinates of the midpoint of the
segment.
The average of two numbers is a
number that is half way between
the two numbers.
æ x + x2 y1 + y2 ö
Midpoint = M = ç 1
,
2 ÷ø
è 2
M
M
B
E
Examples:
Segment AB: A = (-7, 9) and B = (-2, -4)
Distance from A to B =
(-7 - (-2)) 2 + (9 - (-4)) 2 = (-5) 2 + 132 = 194 » 13.93 units
Midpoint M located at:
æ -7 + (-2) 9 + (-4) ö æ -9 5 ö
,
ç
÷ = ç , ÷ = ( -4.5, 2.5 )
2
2 ø è 2 2ø
è
Segment CD: C = (9, 8) and D = (3, -2)
Distance from C to D =
(9 - 3) 2 + (8 - ( -2)) 2 = 6 2 + 10 2 = 136 » 11.66 units
æ 9 + 3 8 + (-2) ö æ 12 6 ö
,
Midpoint M located at: ç
÷ = ç , ÷ = ( 6,3)
2 ø è 2 2ø
è 2
Segment EF: E = (-4, -7) and F = (10, -9)
Distance from E to F =
(-4 - 10)2 + (-7 - (-9)) 2 = (-14) 2 + 22 = 200 » 14.14 units
æ -4 + 10 -7 + (-9) ö æ 6 -16 ö
,
Midpoint M located at: ç
÷=ç ,
÷ = ( 3, -8 )
2
è 2
ø è2 2 ø
D
M
F