Slope / Gradient

The concept of slope is used in various sections of mathematics and worked with
quite often when
solving and graphing linear equations. The slope or degree of slant of a
line is defined as the degree of steepness or incline of the line.

In more mathematical terms, given a plane containing both the x-axis and y-axis,
slope can be defined as change in the y-coordinate divided by change in the x-coordinate.
Slope is usually denoted by m

where the Δ symbol means change in. The change in y is the distance
between both y values, which is also called the rise. The change in x is
the distance between both x values, which is also called the run. The slope
is also known as the rise over run.

Given two points (X1,Y1) and (X2,Y2)

which is the same as

Although it doesn’t matter which point you start with, consistency is a must. Below
is an example of a WRONG way to calculate the slope

whatever point you choose as the starting point in the numerator MUST be the same
point you pick in the denominator

Slope can be positive or negative or zero:

• • Positive slope means that the line is increasing, in other words moving from left
    to right.

• • Negative slope means that the line is decreasing or moving from right to left.

• • Zero slope on the other hand means that the line is horizontal i.e. parallel to
    the x-axis.

In some cases, the slope may be infinite or undefined and this means that the line
is vertical i.e. parallel to the y-axis. This occurs when there is no change in
the x-axis i.e. (X1 – X2 = 0)

The magnitude of the slope shows the steepness of the line; the greater the magnitude
of the line the steeper it is.

Slope Intercept Form

Given a straight line with the slope-intercept form of a line, y = mx + b,
where m represents the slope and b is a constant which is also called
the y-intercept. The y-intercept is defined as the point on the y-axis at which
the line (whose equation is given) cuts the y-axis.

Keeping in mind that at any point on the y-axis the x-coordinate is zero (x = 0),
an easy way to get the y-intercept from the equation of a line y = mx + b
would be to simply set x = 0 such that y = b.

For a given straight line, the slope is consistent along the line so it wouldn’t
matter what points on the line you pick to calculate the slope.

Gradient in Geometry

In geometry, given a line
that makes an angle θ with the x-axis, the slope m is defined as

In geometry, the gradients of a lines can be used to determine their relationship
i.e. whether the lines are
parallel to each other or
perpendicular. For example: Given two lines with slopes m1 and m2

• The two lines are parallel if and only if their slopes are equal (i.e. m1 = m2) and
  they are not coincident (i.e. don’t lie on top of each other) or if they both are
  vertical and therefore have undefined slopes (i.e. m1 = ∞ and m2 = ∞

  

• The two lines are perpendicular if the product of their slopes is -1 (i.e. m1 x
  m2 = -1) or one has a slope of 0 (a horizontal line) and the other has an undefined
  slope (a vertical line) i.e m1 = 0 and m2 = ∞ or m1 = ∞ and m2 = 0.

  

From the above, notice that given two perpendicular lines and the slope of one line,
you can always find the other slope from the relationship

i.e.

Slope in Calculus

Calculus mostly deals with
curves whose slopes/gradients may be harder to compute using the algebraic method.
When dealing with curves, the gradient changes from point to point so we can only
define it at a single point. The gradient at that point is defined as the gradient
of the tangent line to that point. The tangent line is defined as a line to a curve
that only touches one point on the curve.

Given a simple curve y = x^2

The gradient at a given point say (1,1) is found by taking the derivative of the
equation and then substituting for the point i.e.

gradient m at (1,1)

Examples of Slope / Gradient

(1) Find the slope of the line between the points (1,2) and (3,6).

(2) Find the slope of the line   3y = 2x + 1

This equation is not in slope intercept form, so we divide by three to find our
m value.

(3) Find the slope of the line 30 – 2y = -0.5x

Isolate y to put the equation in slope intercept form.

(4) Find the gradient of the given line y = mx + 3 at the point (2,5)

substitute for x and y

such that