What is the simplest way to find the gradient of a slope?
If you are looking at a line graph, the gradient is 'rise over run'. If you count a number of squares up on the vertical axis (y axis) and then count the number the graph has moved along the horizontal axis (x axis, often 'time') in that time (forwards or backwards, depending if the slope is positive or negative), then the gradient is the number of squares up on the vertical axis divided by the number of squares along on the horizontal axis.
If you have the equation of the line graph, which in standard form is
y = ax + b
the number 'a' is the gradient of the slope of the line, and 'b' is the intercept of the line (where the line cuts the vertical or y axis). In other words, the gradient is the coefficient that accompanies the x variable.
On distance-time graphs, the gradient of the line equals the speed (or velocity if there is a direction of travel). if the distance is in km and the time in seconds, then the speed is measured in km/s (distance units divided by time units), which is read as km per second or kms^-1.
Line graphs and beyond:
For more complex graphs in general, the gradient is the rate of change of the graph in the y direction with respect to change in the x direction. The rate of change can be established by differentiating the function of the graph f(x) to get the derivative function f'(x). For linear graphs (of order 1), x to the power 1 differentiates to 1, since the rate of change is constant (every movement on the x axis corresponds to the same amount of change each time on the y axis). For higher order or non-linear functions, for example a parabola, the rate of change is no longer constant. In the case of a parabola (involving an x^2 term), the rate of change increases (or decreases) linearly in x, since the derivative of x^2 is 2x (multiply by the old power and decrease the old power by one to get the new power). Like a lot of maths, rules on differentiation need to be memorized as there are different rules for all the different types of non-linear functions. Some graphs have places where there is no slope or derivative as it takes no value for that particular value of x, or there is a discontinuity, where the graph might shoot of the page, or move in steps.
Gradient (Slope) of a Straight Line
The Gradient (also called Slope) of a straight line shows how steep a straight line is.
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.
Divide the change in height by the change in horizontal distance
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.