Line and Definition of Slope

Game: Line Graphs

Slope of a line (steepness)

Consider a particle moving along a non vertical line segment from a point p1( x1,y1 ) to a point p1( x1,y1 ). The vertical change y2 – y1 is called the rise, and the horizontal change x2 – x1 the run.


Definition

If P(x1, y1) and P(x2, y2) are points on a non-vertical line, then slope m of the line is defined by:


It does not matter which point is called P1 and which one is called P2
Slope of P1P2

= (y2 - y1)/(x2 - x1)

= -(y1 - y2)/[-(x1 - x2)]

= (y1 - y2)/(x1 - x2) = Slope of P1P2


Any two distinct points on a non-vertical line can be used to calculate the slope of the line. To measure the slope, we generally move from left to right when measuring the distance travelling horizontally.
Because of this, sometimes the concept of fall replaces thar of rise!


Example
In each part find the slope of the line through
(A) (6, 2) and (9, 8)
(B) (2, 9) and (4, 3)
(C) (-2, 7) and (5, 7)


Solution:
We know that slope of line through two points P1(x1, y1) and p1(x1, y1) , is given by
m = (y2 - y1)/ (x2 - x1)
So
a) m= (8 - 2)/(9 - 6) = 6/3 = 2
On coordinate plane xy


Similarly
    b) m= (3 - 9)/(4 - 2) = -6/2 = -3
On coordinate plane xy


Also
c) m= (7 -7)/[5 - (-2)] = 0/7 = 0
On coordinate plane xy


      Definition (Angle of inclination)
For a line L is not parallel to x-axis, the angle of inclination is the smallest angle φ measured counter-clockwise from the direction of the positive x-axis to L.
For a line parallel to the x-axis, we take φ = 0
As shown in the following figures.
       


If m is slope of line then,
m = rise/run
    = Rate of change of y with respect to x


Theorem

For a non-vertical line, the slope m and the angle of inclination φ are related by
m = tan φ


Example:
Find the angle of inclination for a line of slope m = 1 and the angle of inclination for a line of slope m = -1


Solution:
If m = 1 tan φ = 1, so that φ = π/4 = 45°

If m = -1 tan φ = -1 since, 0 < φ < π φ = 3π/4 = 135°


Theorem

Let L1 and L2 be lines with slopes m1 and m2, respectively
  (a)   The lines are parallel if and only if m1 = m2
  (b)   The lines are perpendicular if and only if m1m2 = -1


Proof: (a)
If L1 and L2 are non vertical lines, then their angles of inclination φ1 and φ2 are equal.
            φ12
Thus
m1 = tanφ1 = tanφ2 = m2

Conversely, if two slope lines are equal, i.e.
M1 = M2
⇒ tan(φ1) = tan(φ2)
⇒ φ1 = φ2
So, lines are parallel.


(b) Assume that φ1 < φ2
Then referring to the figure
m1 = tanφ1 = c/h

m2 = tanφ2 = -h/c


The proof of the converse is left as an exercise.


Theorem

The vertical line through (a, 0) and the horizontal line through (0, b) are represented, respectively, by the equation
x = a and y = b


Theorem

The line Passing through P1(x1, y1) and having slope m is given be the equation
y - y1 = m(x - x1)
This is called the point-slope form of the line.


Theorem

The line with y-intercept b and slope m is given by the equation
y = mx + b
This is called the slope-intercept form of the line.


Game: Line Graphs


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