What even is a slope?

Question

I imagine slope as the steepness of a line which means that it should probably tell us an angle which the line makes with the positive x axis .But we get a number .WTH does that even mean? What am i suppose to even do with that number? What is it's significance? How do i use it?

Sorry for the rant but it's frustrating to study in an education system which explains absolutely nothing.

Answer 1

"steepness of a line" is a good concept for slope

So, a flat, horizontal line has zero "steepness" --- slope = 0

Other words for slope --- incline, grade, slant, tilt

Slope is the "rise over run" of a line ---- That is, how much does it go up as it goes right?

Note that a line that goes down has a negative slope.

The line y=0 is the x axis. It has a slope of 0. The line y=5 is also horizontal (slope=0).

The line y = x has a slope of 1 because it rises 1 unit for each unit going right.

The line y = -x has a slope of -1.

The slope-intercept form of the equation of a line looks like:: y = mx + b

where m=slope and b is the y-intercept (the place on the line where x=0).

A vertical line rises an infinite amount for a horizontal run. Therefore, its slope is UNDEFINED.

Vertical lines may not be represented in slope-intercept form, they must use the standard form (Ax+By=C): For example, x=3.

Answer 2

Take the simple function y = x. When graphed, it includes the points (x,y) equal to (-2,-2), (0,0), (4,4), and all other possible points where the x and y coordinates are the same number.

The slope of a function is the ratio of the "rise" of the graphed line over the x-axis to its "run" away from the y-axis.

For y = x, the graphed line will rise 1 unit for every unit that it runs from left to right. Going from (7,7) to (20,20), the difference in y-values or rise is (20 - 7) and the difference in x-values or run is (20 - 7) as well. One then obtains the slope of y = x as (20 -7)/(20 - 7) or 13 ÷ 13 equal to 1.

Take then the angle between the line y = x and the x-axis as the arctangent of 1 equal to π/4 radians or 45 degrees.

Further study in Calculus will give the vital concept of the Derivative as the slope of the line tangent to the graph of y = f(x) at (x,y) = (a,b) for all (a,b) on y = f(x). The Derivative of y = x² is 2x. For the point (2,4), the slope of the line tangent to (or touching one point of) y = x² at (2,4) is equal to 2(2) or 4.

Since y = x² is not a straight line but a curve, the slope of each line tangent to y = x² changes for each value of (x,y) that satisfies y = x². The Derivative is used in analysis of problems involving instantaneous rate of change.