Now, where were we with Grant? Oh, that's right, he was on the plane. He's on the plane going from home to the Cxpo in the city Udacity. But Grant is super impatient and he would really love to find out where the halfway point between his hometown and Udacity is. He found out earlier that the exact distance between these 2 points is the square root of 530,000, in miles, of course. So, he just measure half of the square root of 530000 along this line, that doesn't seem like a very exact or very easy to do either. But just by looking at our graph, I'd like you to give me an estimate for the midpoint is. That is, what if a coordinates of the point that's halfway between these 2 points along this line?
The midpoint is located right here. The coordinates of that point are 100 and thought halfway along the line was, and then figure out the coordinates of that point. But is there a way that we can use the coordinates of the two points on the line that we already know to find the coordinates of the point halfway between them? In fact, there is. We know that home is located at 0,0, the origin, and the Udacity is located at 200 700,. Going halfway between these points, means going both halfway in the x-direction and halfway in the y-direction. So, let's look at those coordinates individually. For the x-coordinate, we start at 0 and we want to go halfway to 200, half of the distance between 0 and 200 is 100. So, you just go over that much. That automatically gives us our x-coordinate. The y-coordinate works in pretty much the same way. Our initial y-coordinate of home is 0, we want to go up 700, to where Udacity located vertically. This distance is 700 miles, so half of 700 is
So to get from this expression to this expression, you could either first place your cursor right here just to the left of this x and inside this opening parenthesis, and get rid of the inner set parenthesis first. Or alternatively, you could have started with the outer parenthesis, too. You put your cursor here and then press Delete. And then, of course, go back to the inner one and do the same thing. Order here does not matter as long as you're following this rule.
Remember that MathQuill is the tool that's allowing us to write many of the things in this class really beautifully. So, and we want you to enter things like x to the 4th over 2. It actually looks like this in the spot where you input things. But as great and powerful as MathQuill is, there are a couple of little idiosyncratic things that you should know about how to enter things. Especially when it comes to using parentheses. You may have noticed that if you type one parentheses, one opening parentheses, that is, in a text box where MathQuill is being implemented, then the closing parentheses appears as well. And you'll be typing in this spot in the box in between them. This generally makes your life much easier. But in certain situations, it can be kind of tricky. One of these interesting, and potentially troublesome features, is that once you have a set of parentheses, and then something written inside them, if you want to delete the parentheses but leave whatever you've written inside, then you can't have your cursor sitting out here and then press Delete because that will delete the entire expression. So let's say you'd written 26 m inside parentheses. If you put your cursor right here and press Delete or Backspace, everything would disappear. If instead, however, of having the cursor out here to the right of the entire expression in the parentheses, you put the cursor just inside of opening parentheses, so just to the right of it. Then, you can press Delete or Backspace or whatever your computer uses to get rid of things before the cursor. And that'll leave you with whatever was inside parentheses originally. I'd like you to get some practice in how this works using this expression right here. So once it comes time for you to take this quiz, I'd like you to type in 3 plus open parenthesis 2 x y minus 7 open parenthesis x z, and then two close parenthesis. And then, in as few keystrokes as possible, change this to just 3 plus 2 x y minus 7 x z. So, this is just giving you a chance to figure out how to delete parentheses without deleting expressions that were inside of them.
Now at certain points in the course you may need to enter coordinate pairs using of course typical coordinate notation with parenthesis around the outside of them that contain fractions for the coordinate values. This is actually a little bit tricky to do in mathquill so we are going to talk about who you do this. I'm going to tell you exactly how to type this out. So let's say that we wanted to type. So, let's say that this is the order pair that we want to type. 3 over 4, negative 1 and a half. I'm going to tell you exactly what keys to press. And then what will appear as you type each one of them. So, the first thing we have is this open parentheses. But when you type that, both parentheses appear. The next step is to type a forward slash. At which point, the space inside the two parentheses, will turn into the input for a fraction. So there will be a line, and a space for you to enter the numerator and the denominator. Your cursor at this point will be in the numerator. And as unintuitive as it might seem, what we want to do, is to get the cursor to be right here, inside the first parentheses. But before we have the option to type inside the fraction. So you just need to type a left arrow key to move the cursor over here. Then you can type a comma, which will be the comma between our two coordinates. At this point your cursor will be right here, between the comma and the fraction spot. So you need to type the left arrow key to move it back here. Now we just need to create the first coordinate. Then you need to type a forward slash again. Which will create another spot for a fraction to be input. And then you can use the arrow keys as you normally would between fractions to go to each numerator and denominator and enter the values that you want to. So you can see that whats happening here, in terms of the order that things appeared in over here depending on what I told you to type, that you have to actually create the spot for the Second coordinate as a fraction before you create the first one. If you try to create one fraction and then put a comma, it's not going to work. Another option would be for you to create two separate fraction spots and then move to the spot between the two of them and then put a comma. This is just a way that works really easily for me. But of course, as always, you can experiment and figure out what works best for you. So, thinking about how you create 2 coordinates that are each fractions. I'd like you to try out typing each of these coordinate pairs. First one we have is 3 4ths, negative 1 half. Then we have 1, 2 over 7. And then we have negative 1 8th, 3. So just play around with these, and see if you can get comfortable with this format.
So, the thing about entering these points, is that there are actually a ton of different ways you can do it. I simply wanted to give you one method that you can use that's pretty reliable for creating different fraction slots in ordered pairs. So, for our first point, 3 quarters comma negative 1 half, you could, of course, use this method that I showed you up here. But you could also just enter this from left to right and it should turn out just like this. So, for some ordered pairs involving fractions, things aren't really that complicated. However, this one right here is a pretty difficult one, I think. If you try, for example, to enter to everything in here from left to right, here's what you'll end up with, you open parenthesis and then a 1 comma 2 as a denominator over 7. That's definitely not what we want. So here, I think it's easiest if you first enter the fraction 2 over 7 then go backwards and write the 1 and the comma. Negative 1 over 8 comma 3 is another one that's pretty simple to enter. You can also do this from left to right, or you can use the method of inserting fractions that's where you need them first. So, the moral of the story is, when your typing things using MathQuill, you always need to be super careful to check what you're typing and make sure that it's actually what you want to be typing. Also, the more practice the better. So, just keep trying things like this out, any chance you get.
What if I give you two random points, 3, 15 and 10, negative 2. Can you find the halfway point between them, the midpoint?
The midpoint between 3,15 and 10, negative 2 is the point 6.5, 6.5. Of course, you could write this as a fraction instead of a decimal too, if you wanted to. But how do we get that? Well, in order to get halfway from here to here, we know that we need to go half as far both horizontally and vertically. Let's once again start with the horizontal distance, or the distance in x. Our x coordinates or our given points are 3 and 10. So halfway between 3 and 10 is what? Well, the distance from 3 to 10 is just 10 minus 3. So that means that halfway between 10 and 3 is just 10 minus 3 over 2. And simplifying that gives us 7 halfs, or 3 and a half. This, however, is not our coordinate. This is just the distance that we need to move from one point toward the other point. So to figure out our actual coordinate, we need to start at one of our points and walk toward the other point this far. So our x coordinate is just equal to 3 plus 7 halves, which is just 13 over 2, or 6.5. Let's do the same thing for the y coordinate. Our y coordinates are 15 and negative 2. If we use our handy dandy grid here to count how far this distance actually is, we can see that it is 2, move 17 boxes vertically to get from negative 2 to 15. Half of that is just 17 over 2 or 8 and a half. But again, this is not actually our Y coordinate. This is just the distance vertically between these 2 points. To find the coordinate, we need to start at one of these points, and then move this vertical distance toward the other. So if we start at 15, we need to move down 8 and a half, since negative 2 is below 15. And this gives us a y coordinate of 13 over 2. Let's graph this point just as a sanity check. 6 point 5 6 point 5 should be right about here. And if we draw a line from our first point to our second point, we see that this point does in fact lie on the line, and looks half way between them.
This might be a little bit tricky, but what is the midpoint between 2 generic points with coordinates a,b and c,d? Please try to simplify each of your coordinates in your answer, so each one is just a single fraction.
Our answer, interestingly enough is, a plus c over 2, b plus d over 2. To figure out how we got to this final answer, lets start again with the x coordinate of our midpoint. Our x coordinates of the two original points are a and c. And lets say they want to start at a and move halfway towards c. First we find half the distance between these two coordinates. Which is the distance, c minus a, just divided by 2. Then you need to shift that over graphically so that we're starting at point a, instead of starting at the origin. So we need to tack on a distance in x of a. I said that I wanted your answer as a single fraction, so we need to simplify this a little bit. Putting this all over one common denominator gives us c minus a plus 2a, all over 2. And in the end we get c plus a over 2, Or of course because of commutivity, a plus c over two. We do the same exact thing for our y co-ordinate except of course we use the y values of these two points. And we end up with d plus b over 2. Together, these two coordinates give us our final midpoint.
Thinking about our answer to the last quiz for a general way of writing the midpoint between any two points, what do you think is a good way to sum up what we're actually doing when we find that? To find the midpoint between two points, do we find half the distance in both directions? Do we add the corresponding coordinates, the x-coordinates and the y-coordinates of the two original points together? Do we take the average of those corresponding coordinates, or do we just find the distance between the two points?
When we find the midpoint between two points, we're really just taking the average of their x-coordinates and their y-coordinates. In the quiz before this one, we found that the midpoint between points a,b and cd is just a plus c over of a and c. And the same is true of the y-coordinate for b and d. So, thinking of finding midpoint this way, as just averaging x and y coordinates, sometimes makes the math feel a little more intuitive and easier to remember how to do.
Now that we know that finding a midpoint is as simple as averaging x coordinates and y coordinates, let's go back to this quadrilateral we, we are looking at earlier. Can you find the midpoint of each side? Don't forget to write these as points, so with coordinates
Here are the midpoints that I found using our averaging method. So just like you saw in the previous quiz, all we need to do to find each midpoint is take the average of the x coordinates of the points that lies between, and then also, the average of the y coordinates of the points that lies between.
Since you just calculated the midpoints of each of the sides of this quadrilateral, I've now plotted them for you on our coordinate plane. I think you'll find something pretty interesting though. What happens if I connect all these midpoints together? Hm, we get 8 different quadrilateral. Think for a second about these line segments joining these points and the shape that we formed. Here's a box for you to just record your thoughts about our new shape.
Thank you for jotting down your notes. I just wanted to get your brain jogging before we do some calculations, and figure out how to quantify what's special about this black figure here.
Well, since we've been talking about distances a lot, we might as well find the length of each side of this new black quadrilateral. This time, however, we're just going to use these coordinates of the points that we know. So, what did we use before? Well, we used the distance between the x-coordinates, and the distance between the y-coordinates, squared each of those distances, added them together, and then found the positive square root of that quantity. Interesting. So, you can formalize this process for any two points, where one of the points has the coordinates that I'll call x1 y1,. And the other point has coordinates x2 and y2. I want to use these subscripts here because that lets me still show that these two coordinates are in fact x-coordinates, and these ones are y-coordinates, but, that they belong to different points. Now that we have this handy formula, for the distance between any two points, we can calculate distances just knowing coordinates. That means, we don't need to rely on our picture anymore. Although, of course, it will be helpful in the future to go back and reference it. We know that we can use this formula to calculate the distance between any two points, so let's try it for each of the points that creates the outline of our black quadrilateral, on the visual that I've now conveniently taken away from you. The notation here that we're using is similar to what we did before. MN, for example, is the distance between M and N, two points on our list. You can consider this just two letters that form a single variable. So, NP is the distance between N and P, PQ is the distance between P and Q, and so on.
Let's start out by calculating the distance between M and N, or the distance that we're just calling MN. I'm going to take the square root of a bunch of stuff. Our x-coordinates are 11 half and 2, and when you take the distance between them, then we square that. We add to this the same thing for the y-coordinates. And then, we just simplify. After some more simplification, our final answer is the square root of 218 divided by 2. Interestingly enough, when we calculate the other's three side lengths, we get that PQ and MN have the same length, and also that NP and QM m have the same length. Let's keep this in mind for discussion slightly later on.
Let's analyse this formula, the distance formula, for a second. If I switch the order of x2 and x1 in this formula or of y2 and y1 in this formula, is that going to change what I end up getting?
And the answer is no. Since each of these distances is going to be squared, it doesn't matter which order we have the coordinates in. If instead we had x1 minus x2 and then wanted to square that, well, this is the same as negative 1 times x2 minus x1. And if we square all of that, we end up getting back to x2 minus x1, the whole quantity squared since when we squared negative 1, we get a positive 1. So, it doesn't matter which point you start with and which point you end with when you're finding the distance between two points.
We just found out that the opposite sides of our quadrilateral in black here, are the same length. MN is equal to PQ and NP is equal to QM. But is there anything else that is special about this black shape? You may remember from Geometry that any quadrilateral whose opposite sides are equal to one another is called a parallelogram. So, the opposite sides of a parallelogram are congruent to one another or of equal length, but they are also something else. They are parallel to one another. But how can we tell they're parallel, mathematically? And what does being parallel really mean? That's what we're going to look at next.
Now that we know that m, n, p, q is a parallelogram, and that its opposite sides are parallel to one another, what do you think that the term parallel refers to? Are two things that are parallel of equal length? Do they have the same slope? Is the way that they relate change in y to change in x the same? Or do they intersect the same segments?
It looks to me like the slope of these opposite sides is the same. Slope, however, as we'll soon talk about, has to do with how the change in y and the change in x are related. Any two lines that have the same slope are parallel. They will never touch one another. They'll always just be the same distance apart along their entire length.
So, what really is slope? Well, here's a line connecting two points, 2,3 and you think a good general definition of slope would be? Is the slope of a line how many units to the right are moved for every unit that the line goes up? Is it how many units the line moves up for every unit it moves to the right? Is it the number of units moved vertically plus the number of units moved horizontally? Or, is it lastly, the number of units moved vertically minus the number of units moved horizontally?
The slope is how many units a line shifts upward for every single unit that it moves to the right. For the segment joining these two points right here, you see that as we move from one point to the other, we're translating 1, 2, 3, 4, 5 horizontally to move five to the right, and then up, 1, 2, 3. The ratio of this vertical distance traveled versus this horizontal distance travelled is 3 over the right, we can see that we move up by this distance, which is 3 5th of 1 unit.
Earlier on, we looked at the graph of y equals 3 x minus 5. We saw that we got this line. Thinking about your new understanding of what the slope of a line is, can you tell me how many units up, for every unit across to the right, this line goes? It can be helpful to start at a point on the line and draw a right triangle, first moving out to the right and then up, to help you figure this out.
This line goes 3 units up for every unit that it goes to the right. If we look at this point right here, for example, and we step 1 unit to the right, we need to count up 1, 2, 3 units before we hit the graph again. We can draw a right triangle going over 1 and up 3. This then, is the slope of this line.
As a quick check that you know how to find the slope of a line from its graph, please tell what the slope of this line is?
The answer is 5 over 2. Let's say I want to start at this point right here, the point 0,1. I can find another point that lies on this line that's easy to pick out, like this one right here, point 2,6 and figure out that we count over one, two, to the right, and then up from there, one, two, three, four, five. So we take five and divide it by two and that gives us the slope. We can also see that if we wanted to step over one to the right, we would need to step up 2 and a half units to be back on the line and 2 and a half is equivalent to 5 halves.
We know one way to articulate what we do to find the slope of a line, but there are definitely other ways that we can think about this as well. Please pick the best one of these choices. When we try to find the slope of a line, do we divide the x coordinate by the y coordinate, divide the increase in x by the increase in y, divide the increase in y by the increase in x, or multiply the increase in y by the increase in x?
The answer is this third one. We notice here that we had to move up five units everytime we moved over two units, and since our slope is 5 halves, we're dividing 5, the increase in y, by 2 which is the increase in x.
We just said that the slope of a line is equal to its increase in the y direction divided by its increase in the x direction. Here are two new points that I've drawn for us, 1,7 and 5,3 and I've drawn the line that runs through both of them. Notice that instead of pointing up and to the right, this line is pointing down to the right and this affects the slope of this line in some way. Thinking about that, what can we say about the slope of this line? Is it a fraction, is it positive, is it negative, is it not defined, or is it zero?
The slope of this line right here is negative. Notice that slope as we defined it, is the increase in y divided by the increase in x. But as x increases over here on our graph, y actually steps downward. We can't draw a triangle like we did before, or rather, we can draw a triangle, just a different triangle. We have to notice that as we walk over to the right, we walk down instead of up. This means that our slope is negative instead of positive.
Now that we talked a little bit about what this slope might be like, let's actually find it. Please first tell me what the increase in x between these two points is, and also the increase in y, and then from that, figure out the slope.
As we move from the point 1,7 to the point 5,3, our x-coordinate moves to the right, 1, 2, 3, 4. This is the positive direction so we can say that our increase in x is positive 4. Now y, on the other hand, as we move from here to here, has to step down, 1, 2, 3, 4. So, we actually have a decrease of 4 and y. A decrease is the same as a negative increase. So, our increase in y is negative our slope is negative 4 over 4 or negative 1.
Let's look at a general situation now, then, for a point a,b and a point with coordinates c,d Now, can you use these coordinate names to tell me what the slope of this line is?
We already know that, in general, the slope of a line is the increase in y for every increase in x. So, the change in y here is going to be d minus b, and the change in x is going to be c minus a. So that is a general answer for how to find the slope of a line between any two points.
When answering the last quiz, i called these points very different letter names. But when we look at this formula for slope, it's not really clear how d and b, or c and a are related to one another. I think that one thing that might help us is if we renamed the coordinates of these points, so that they indicated what each coordinate actually is . Instead of A,B I'm going to call this coordinate x1, y1, and this point is going to become x2,y2. Using this new notation, we can change our formula for slope. I like this form much better. To me it's way more clear now that this quantity on top is the difference between two y coordinates, and this one in the denominator is the difference between our two x coordinates. Now note that the y coordinate of the second point comes first in the numerator, and the x coordinate, the second point, comes first in the denominator. The question I have for you now, is, does it matter which point, when I find in the slope between points I call x1, y1, and which one, I call x2, y2. In other words, if I switch the position of the ones and twos in this equation for slope, would I get the same slope as my answer?
And the answer is no. It does not matter which point we use in which slot in our equation provided that we are consistent about the coordinates of a single point being the same position in both the numerator and the denominator. You can imagine that if we started out instead with y1 minus y2 over x1 minus x2, it's actually pretty easy to show that this is equal to our original equation for slope. We just pull out a factor of a negative 1 from both the numerator and the denominator. You can see that those factors would cancel one another out. Well, you'll be left with y2 minus y1 over x2 minus x1, which was just our otriginal equation for slope.
Here, I have 2 point, 3,2 and 7,10. Now, I've drawn a line between them and I'd like you to tell me just like you have before, what the slope of this line segment is. Try calculating this, first, using the coordinates of the top point, calculating a second way using these coordinates as the ones that come first in the denominator and in the numerator.
The first way I'm going to try to do this is to start with the coordinates of this point. So, I have this y value, 10 minus the y value of this point, which is 2, and I divide that by the x value of the top point minus the x value of the lower point. Simplifying, just gives me 2. Now, let's try starting with the coordinates from the lower point first. We'll take that y value, 2, and subtract doesn't matter which points coordinates we put first, as long as we're consistent.
Here are a bunch of lines that I've drawn on one coordinate plane. You'll notice that I've labeled each of these lines with a name. We have v, w, x, y, and z. And I'd like you to pick which of these categories each of these lines falls into. For some of these lines, there may be more than one category.
But before I elaborate on any of these answers in our grid, let's remember what the general formula for slope is. We can find two points along any of these lines, and then compare them using this formula to find the slope. The first line, v, is a really interesting case. If we pick any two points along it, let's say negative 7,3 and negative 3,3. You'll notice that they have the same y-coordinate, they have the same vertical position. That means that our numerator of our slope formula is going to be zero, since we'll just have 3 minus 3. Zero divided by any number except zero is just zero. So, the slope of v is just zero. w is another really interesting case. It's this vertical line right here. Although, the points along w have a wide variety of y values, they all have exactly the same x value. x equals negative 5 for any point we could place along this line. So, let's just pick 2 again. Maybe this point, negative over negative 5 plus 5. But, this quantity on the bottom is going to give us zero and we know that we can't divide by zero. That is just not allowed, at least not for right now. So, what we say is that the slope is undefined. Line x is pointing up to the right, so its slope is positive. And if we step over one to the right, we have to step up more than one to rejoin our line. So, its slope is not a fraction. It's greater than one. The line y also has a positive slope. But, if we pick a point and move over one to the right, we have to move less than one unit up to rejoin the graph. So, this slope is a fraction. Line z points down to the right, so as we saw before, its slope is negative.
I think it's worth looking into vertical and horizontal lines a little bit more, just because we haven't really seen them that much yet. Please tell me which description matches the vertical line and which description matches horizontal lines.
Let's start with our vertical line and just pick 2 random points along it. So, here's a point, and here's a point. The coordinates of this line point are 2,7. And this is the point 2, negative 3. The coordinate these 2 points on the vertical line share is the x-coordinate. So I guess, all points on a vertical line have the same x-coordinate. Let's do the same thing for a horizontal line. We have the point negative 8, negative 5, and another point 5, negative 5. These two points, along the horizontal line, share their y value. So, I guess, all points along horizontal lines have the same y value.
Think about what conclusions we drew about the coordinates of vertical and horizontal lines in the last quiz. Try to use that information to figure out what equation describes all the points along line a, and what equation describes all the points along line b. This is definitely tricky, so if you have trouble with it, no big deal at all.
The equation for line A is x equals 2, and the equation for line B is y equals negative 5. Let's talk about why. Remember that along any vertical line, every single point has the same x value. The y values are going to range everywhere negative infinity all the way to positive infinity. But the x one it stays constant along that entire range. So, this is the only equation that we can write that fits every point along that line. There's a similar deal for line B and other horizontal lines. But regarding their y-coordinates instead of their x-coordinates. We said that every point along a horizontal line has the same y value. Its x values go from negative infinity to positive infinity. But the only equation that accurately describes every point along the line is y equals negative 5. Vertical and horizontal lines then have sort of special equations because they only have one variable in them. This is pretty different from equations of other lines we've seen, like for example y equals 3 x minus 2. Here we have an x and y in the same equation. What these equations show is, respectively, that for vertical lines, this relation with x is going to be true regardless of y value. And for horizontal lines, this equation involving y is going to be true regardless of x value, or for any x value.
Last time, we heard from Grant, a long, long time ago, he was on the plane going from his hometown to Udacity where the glasses expo was. He made a map of his flight path. We found its length, we found the midpoint between these 2 points, so on and so forth. But now, we've been talking about slope, I would like you to tell me the slope of this line.
From earlier we developed this general equation where we use the coordinates of home are (0,0) and the coordinates of the city Udacity on the map are (200,700), not forgetting the scale of either axis. I'm going to pick the coordinates of this point to be x2 and y2, and this one to be x1 and y1. Let's plug them in. We have 700 minus 0 over 200 minus 0. Well that's pretty simple. It's just 700 over line is 7 over 2. Since Garrett really wants to know how to find out the coordinates of any point along this route, we'll eventually want to incorporate this slope into an equation for this line, but we're not there quite yet. Great job on this really difficult lesson.