Rate and Change of Slope A1711 Activity Introduction Hey there, I m (name) and today I m gonna talk to you about rate of change and slope. Slope is the steepness of a line and is represented by the letter m. Video 1 - Introduction While you re watching the video, be sure to pay close attention to the method that s being used to determine if a set of ordered pairs is a linear function. Remember, a linear function has a constant rate of change, or slope. Let s take a look. Video 1 Okay, so now we re going to take a look at a set of ordered pairs and try to determine if that ordered pair satisfies a linear function. Remember that in order for it to satisfy a linear function, it means it has to have a constant rate of change. And by a rate of change you re referring to a slope or the ratio of the vertical change to the horizontal change. We also talk about in terms of the delta y over delta x. So how we re going to approach this problem? Well, the first thing we can do is take a set or ordered pairs and actually put it into a table by remembering that each order pair has an x value and a y value. So we can take all the x value from the order pairs and put them in one column and now all of our y
values we put them in another column. If we do that we get something that looks a little bit something like this. (0,0,(1,5),(2,10),(3,15) x y 0 0 1 5 2 10 3 15 So from this table we want to see if can determine whether the rate of change between any two points is going to be constant. To do that we ll take a look at specifically between each points our y and our x. So between our first two points (0,0) and (1,5) we have an increase in our y values of a +5. From 5 to 10 we also have an increase of 5 and from 10 to 15, an increase of 5. So between each of those points our change in y stayed the same. It was constant. So y is 5. Well let s check the x values. Between our first two we re going up 1; between 1 and 2 we re going up 1 and between 2 and 3 we re going up 1 again. So again we see that between these two points the increase is only 1. So it s not changing which means it s constant and our x is 1. Well 5 1 tells us that we have a rate of change or a slope of 5 and we can see that because it s not changing, it s actually constant. And that tells us that we do indeed have a linear function with the constant rate of change. Well, let s take a look not at that table but of a graph of the same order of pairs and if we can prove to ourselves in looking at the graph that it is linear. Again so we have (0,0),
(1,5), (2,10), and (3,15), the same set of ordered pairs we re looking at in our table. Well, now let s see because that rate of change tells us that change in y over the change in x. Let s look at that in terms of the points on the graph. So starting at (0,0) in order to get to the next point we have to go up 5 and to the right 1. And if we go up 5 to the right 1 again we re still on the line. So the rate of change is 5 over 1, 5 over 1 and it s not changing at any point on the line. So our line is going to be straight. It is not curving because there s a constant rate of change and once again we know this means we have a linear function. So we took a look at an ordered set of pairs, put it into a table and then looked at them as a graph and all three times we saw that we do have a constant rate of change which means we do indeed have a linear function. Great job. Video 1 Recap What you want to keep in mind here is that when you re using ordered pairs, the slope of a line can be found by using the ratio of the rate of vertical change to the rate of horizontal change, or delta y over delta x. Video 2 - Introduction Finding slope from a line instead of a set of ordered pairs is still done by using rise over run, or the ratio of the vertical change to the horizontal change. Take a look. Video 2 Okay, so we will take a look at how to find the slope of a set of order of pairs. Now let s see if we can apply that same concept if we have an actual line and not the order of pairs.
So the first thing we need to do is remember that slope is the vertical change over the horizontal change or the change in y over change in x or the rise over run. So in this case you want to figure out where we re going to rise from and where we are running to, we need to pick two points on the line. Now, I like to make things easier on myself. So, I'm going to take this point (0,-4) which is actually our y intercept in this point. Our x intercept, which is (6,0). Well, in order to get from my first point to the second point, I have to go up four units and to the right, six units. What this tells me is that my y, my delta y or my rise is a +4 and my delta x or my run is a +6. Well, 4 over 6 can be simplified. So let s rewrite that as. From the slope of that line is. Now the slope of this line is. We got it with the two points we choose. And if you don t believe me, let me prove it to you by choosing two different points. Actually I'm still going to keep my y because that s the easy one. So it s going to start (0,-4) but this time I want to see how do I have to rise and run in order to get to (3,-2). And in this case, I'm already going up two units and to the right three units. So again, my delta y over delta x equals. So the slope of that line is indeed. Okay, I ll show you another one.
And in this case, you can see that the line it looks a little bit different from the first one because it slanted in the opposite direction. Let s see if that has an effect on the value we get for the slope. Again, the first thing that I want to do is pick two points in the line to determine where we re going to run to and where we re rising from. I'm going to choose, (-2,2) and (8,-3). In this case, my change in y, my vertical change is not going up but I'm going to be going down specifically, I want to go down five units. We represent that going down with a -5. So my delta y in this case is -5 and I still want to go to the right and in this case I'm moving to the right 10 units. So I have a +10. So in this case, my which we can simplify to a. Again, if you don t believe it s a slope with the same between any two points, let s test it, by going between our X and Y intercepts. My y intercept is here at (0,1), and my X intercept is here at (2,0). So in this case, to move between those two points my change in y is a -1 because I am going down 1. My change in x is a +2 because I'm moving to the right 2. So a again is a. So the slope of this line is the same. And because that slant on the line was negative, we know that we re going to have a negative slope, okay.
Two more examples. The first one, we have a horizontal line but we re going to do the same as that thing upon the slope, find two points and move between them. So I'm going to pick (-2,3), and (2,3). How do I get between these two points? Well, I need to rise, all right, I don t rise at all. So in this case the delta y is going to be 0 but I am going to have to move to the right, one, two, three, four units, so my delta x is a +4. Well, 0 over 4 simplifies 0. So the slope of this line is 0 and actually anytime we have a horizontal line, we are never going to go up or down, so our rise is always going to be 0 and therefore the slope of the line is going to be 0 also. So we have a horizontal line, we need a vertical line too. So let s try this one as our last example. The two points I'm going to choose this time are (-2,-3) and (-2,+3). So to get from one point to the next, I have to rise six units from my delta y is a +6 and I have to run. Oh! I'm not running at all. So I m going to have 0 as my change in x. But we know in math that we can t really divide by 0, so the slope of this line is actually undefined. And indeed anytime we have a vertical line, the slope is going to be undefined because we re never going to have a run, we re never going to move left or right. So we looked at positive slopes, negative slopes, zero slopes and slopes that are undefined. So now you get to go practice this on your own. Good luck. Video 2 - Recap
If you have a line, you can determine the slope by comparing delta y to delta x, or rise over run, just like this. You can also find the slope of a line by selecting any two points and counting the number of units in the rise and the run, then writing the ratio of the value of the rise to the value of the run. Remember, the value of the slope determines the slant of the line. End of Activity Review Remember, the slope, or the steepness, of a line can be found using any two points on the line. The slope of a line is the ratio of delta-y to delta-x. Delta-y is the vertical change and delta-x is the horizontal change.