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Have you ever ridden a bike uphill or skied downhill?
Then you sure know what a slope is.
In this video, we’ll focus on comprehending what a slope is, and learn to find the measure of a slope on a Cartesian plane and identify the different types of slopes.
But before we do that, let’s take a look at
the ramp,
the pyramid,
and the slide.
All these are examples of slopes.
Mathematically speaking, a slope or gradient is a number that describes the steepness of a line.
Now imagine you are riding a bike uphill.
Let’s transfer that onto a Cartesian plane.
This line here is our hill.
As you move from the left to the right on the x-axis, the line goes up.
Let’s figure out how steep this hill is.
We'll pick any two points on this line.
And name them point A (3, 2). And point B (6, 5).
Starting at point A we'll move up vertically until we are in line with the point B.
This vertical change in y is called the rise.
It is important to pay attention to the direction we move.
Since, we had to move up 3 units to be in line with point B, the rise is a positive 3 units.
From here, we'll move horizontally toward the right until we reach point B.
This is known as the run.
Moving toward the right indicates the run is positive.
We have our run or change in x which is a positive 3 units.
The slope represented by the letter m is the rise over run.
That is the change in y over the change in x.
The slope m here is 3 over 3 and that equals to 1.
A positive slope indicates an uphill movement.
How about choosing a different set of points on the same line?
Let’s call them point C(2, 1). And point D(4, 3)
Starting at point C, we’ll move 2 units up and 2 units across to reach point D.
Slope is how much you move up over how much you move across.
That is 2 over 2.
And our slope is 1.
From this, we can conclude that the slope of a straight line is constant.
You can actually pick any two points on the line and the slope will still remain the same.
Let’s move on to our next slope.
Imagine this snowboarder sliding downhill.
Let’s represent that on our Cartesian plane.
As you move from the left to the right on the x-axis, the line goes down.
We’ll pick two points on the line: point A (1, 3). And point B (2, 0).
Starting at point A, we’ll go down until we are in line with point B.
Now, that is 3 units down.
As we are going down our change in y is gonna be negative.
So, that’s a negative 3 units.
Now, let’s move across to reach the point B.
That’s just 1 unit.
The change in x is obviously positive, as we have moved toward the right.
Slope is the change in y over change in x
We have a negative 3 over 1.
That’s a negative 3.
A negative slope indicates a downhill movement.
Now imagine driving on a flat road.
The road looks like this horizontal line that has no steepness.
The run is 4.
But there is no change in y, so our rise is 0.
Slope is equal to rise over run.
That’s a 0 over 4.
The slope in this case is zero.
Let's move from no steepness to the steepest line.
The rise is 5 units.
But there's absolutely no movement across. Hence, the run here is 0.
Slope is the rise over run.
We can’t divide 5 by 0.
So, slope of such a line is undefined.
The measure of the slope determines its steepness.
The greater the measure the steeper the slope is.
Let’s take a quick look at the different types of slopes,
starting with the zero or flat slope indicated by this horizontal line.
The line is increasing, if it goes up as you move from left to right.
It indicates a positive slope with an uphill movement.
The measure of the slope m is greater than 0.
And when the line goes vertically up or down, the slope is undefined.
The line is decreasing, if it goes down as you move from left to right.
It indicates a negative slope with a downhill movement.
The value of m is less than 0.
Now, it’s your turn.
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