Remember Grant's chart that we saw for the first few days of glasses wiper production? We came up with an equation in the last lesson to describe this wiper manufacturing situation. If we call total number of wipers produced n and the number of days past d, can you write that equation again for me?
As we figured out before, n equal 23d. With each day that passes, 23 new wipers will be added to the count of the total produced. So, as d increases by 1, n increases by 23.
We figured out this equation in the last lesson and in the previous quiz by finding a pattern in this data right here that relates n to d. But in order to find patterns as easily as possible, we need to see the connection between data like we have in the table, mathematical descriptions like our equation right here, and to add to our list, visual representations, or graphs. For the situation we're dealing with, a graph might look something like this, but how do we make this? Well, we start by plotting points. Each row of data in our chart right here is a pair of numbers, one for d and one for n, and we plot those pairs of numbers on our graph. For the first point I would graph, I have the numbers one and 23. So if we're on our coordinate plane here, on the d-axis which is the horizontal one right here. I'd start at zero and move over one, and then, move up 23 on the n-axis, which is the vertical one. I could do the same thing for 2, 46. I count over from zero in the positive direction, two slots, one, two, and then up, 46.
For pairs of sets of data, like we have in this situation, there's a special sort of notation that we use to help us get ready to graph. So, in this column, I have a set of values that d could equal. And then, in the column next to it, I have a set of values that n will equal, whenever d equals the values to the left. To show which values in either column fit together, I write them inside parentheses together with a comma in between. The reason this helps us with the graphing is that traditionally, the number that comes first corresponds to the horizontal value, or the horizontal coordinate of the point that we're going to draw. And the number that comes next corresponds to the vertical value of the point, or the vertical coordinate. So, thinking about how ordered pairs translate into graphing, where does the point 5,115 belong?
To plot 5,115, we start with the horizontal coordinate. Beginning at 0, we count to the right 5. And then, we deal with the vertical coordinate. From there, we count up 115. And that brings us to this point. But from bringing back our graph, we can see another way that we could have figured this out. By looking at the line that we drew for the entire equation. Remember that on our table, we saw 5 and 115 as a pair of values for d and n that belong together. Since it was from that table that we developed our equation, and from our equation that we got to this graph, it makes sense the graph would fit the table. What's really cool about our line, though, is that our table only had the points that are highlighted here in orange. Those are all the data we were given. But now, with this whole line, we can start to make some extrapolation about data later on.
Using this graph, how many glasses wiper sets will have been produced by the end of seven days?
Since days are represented on the horizontal axis on this graph, what this question is really asking is what vertical coordinate goes with the horizontal coordinate of 7 on this graph. So we start by counting over 7 horizontally, and then moving up to find the point on the graph that has a horizontal coordinate of 7. Then we just need to see what vertical coordinate this corresponds to. This is right above the line representing 160 for n, so that means the answer must be 161. Remember to be super careful about which axis represents which variable. What's even more interesting is comparing this to our equation. If we let d equal 7 here, or just substitute in 7 for d, and then solve for n, we get graph are just different ways of describing the same thing, so of course, they have to agree.
One thing to notice about the last equation we had was that it had not just one variable but 2, n and d. Now thinking about the situation in which we develop this equation, what do you think we can say about what an equation with two variables can do? Pick as many as these answer down here as you think makes sense.
As it turns out, all of these are correct. Equations with two variables can be used to describe a number of situations, and depending on what concepts or items we're dealing with, any or all these could be true. The more two variable equations we see, and you'll be seeing a lot of them, the stronger your intuition for dealing with them will become.
In the last lesson we dealt with Grant's monthly expenses. We found out that he pays $1,200 each month in rent and that every wipers that he bills costs him $6.5o cents to make. Let's suppose that this time we don't know how much money Grant spends total, and instead, we're going to represent that with a variable y. Let's also let x equal the total number of wiper sets he makes. So using this information, can you write an equation that relates y to x?
We can start out by writing y and equals sign, since we know that we want an equation for the total amount of money that Grant spends a month, and that's y. We also know that no matter how productive his business is, he's going to pay $1200 in rent. And this is just a constant amount that he's going to spend, regardless how much else he spends on other things. Things. So, you can write 1,200 as a constant and then a plus sign. He'll spend $6.50 on every single wiper blade he makes. So, we need to add 6.5x to the end of our equation. Voila, we have and equation with 2 variables.
Now what can we do with this equation? Well, we can find out how Grant's monthly expenses change when he produces a different number of wiper sets. By plugging in values of x, and then solving for y. But remember, I'm really lazy, and that would take a lot of time and effort. There has to be a better way. As we saw earlier, representing the data or an equation so we can just see the answer is a great way to have a ton of x and y values right in front of our eyes. So, which of these four graphs do you think might correctly represent this equation that we just came up with? This is only the very beginning of our discussion of graphs, so just take a guess.
As labelled on each graph, the horizontal axis represents the x values, and the vertical axis represents the y values. From our equation, we know that even if no wipers were made or x equals 0, Grant would still have to spend $1,200, which would make y equal 100. That gives us the point 0,1200. x equals zero along the y-axis here. We need to find the graph that intersects the y-axis at y equals makes sense as well. But every pair of glasses wipers that Grant makes, he spends more money because our x value increases. From the equation, we can see that as x gets bigger, y also gets bigger. So, the graph needs to move up and to the right.
To draw the last graph or the last several graphs I had to use a set of axis. One for each variable we have in our equation. The standard way to set this up is to label the horizontal axis the x axis. This demarcates how are x values change. And the vertical axis is the y axis, showing how y values change. Now, every point on this plane is associated with both a value in the horizontal direction, an x value, and a value in the vertical direction, a y value. I also mentioned earlier that we identify points on our coordinate plane with a special notation. The first number listed is our x-coordinate, or our horizontal position. And the second number in the ordered pair is the y-coordinate or the vertical position of the point. The areas that you see on the ends of each axis represent continuation, so the x axis continues all the way out to infinity and also in the other direction, all the way out to negative infinity. And the same is true of the y axis so this coordinate plane are really able to represent, if we want to take as much time and paper as possible, any combination of 2 real numbers. And then of course we can depict it graphically. There is 1 single point on this graph however, 1 point out of that infinite number of points, that's really special. And we call it the origin. Which point of all these ones highlighted in teal do you think is special enough to have such an illustrious title as the origin? Click where you think it is.
The origin is the point 0,0, which is right here, right in the center of our axis. And, of course, we can write it as an ordered pair as well. Both the x-coordinate and the y-coordinate are equal to 0. We saw earlier that this is the point from which we started counting our other coordinates. So, this is sort of like homebase. It's our reference point for figuring out distances to all other points on our plane.
Now, let's use your understanding of how ordered pairs are related to the x and y axes. Please find each of these points on this graph over here, and then in the box next to the ordered pair, fill in the letter that corresponds to the point that you identified it with.
The first thing that's important to remember is that the first coordinate we have in any ordered pair is the horizontal position of the point, and the second coordinate is the vertical position. So, let's start with the first point as an example. We have an x coordinate of 1. So we move from the origin over to the right in a positive direction 1, and a y-coordinate of 2, so we move up two, that's point b. For the second one, we have negative 2 as our x value so starting at the origin, we count to the left in the negative direction 1, 2. And then, the y value is 1 so we count up one. So, that's point d. And we can do the same process for every other point on here. So here, so here are the final answers, luckily none of my letters are the same in any of the boxes cause that would indicate that we had made a mistake. Every pair has a unique position on the coordinate plane. And in the same way, every position on the coordinate plane has a unique ordered pair
This time, I've drawn 4 points on our coordinate plane and I've labelled each one of them with a letter. Next to each letter down here, I'd like you to write in the coordinates for the point that it corresponds to.
For any of these points I'm going to start at the origin, count over in either direction, as many slots as I need to horizontally. And then, count either up or down. So it's going to give me first the x value, and then the y value of that point. So for point a, I need to count over 1, 2, 3, 4. I can see that this is the vertical line that a lies on, and then I need to come up this line toward a, b, starting at the origin. I count in the negative direction, 1, 2, 3, 4, 6, 6. So my x coordinate is negative 6. And then from there, I count up to b, 1, 2, 3, the origin, 1, 2, 3, 4. So negative 4. And then c has a negative y value, since it's below the x axis. So from here I need to count down negative 1, negative 2, negative 3, negative 4. And last but not least, d has a positive x value, 1, 2,
In preparation for his upcoming glasses expo, Grant needs to figure out how to ship all of his wipers and the glasses that they're being modeled on. Unfortunately, he's not going to have the nozzles ready in time but his investors are still confident that they'll be plenty of excitement about the wipers on their own. Grant's just worried about how to get the wipers to the expo. He's planning on bringing all of his model glasses wipers to the expo in a suitcase, but his concern is that the suitcase will be overweight. When his suitcase is empty it weighs 7 pounds, and each glasses wiper set, along with its protective packaging, only weighs 1 half a pound, or 0.5 pounds. Can you find a relationship between the number of glasses wiper sets that Grant packs and the total weight of his suitcase? Please use x for the number of wiper sets and y for the total weight.
We want to end up with an equation for the total weight of the suitcase. And we know that, that is y. So, I'll start by writing y equals something, something that we need to figure out. Well, the total weight needs to equal the weight of the suitcase itself, and then also the weight of the glasses wipers that Grant puts in it. However, we know that the weight of the empty suitcase is just 7 pounds. So, we can just substitute in 7 in that slot in the equation. The total weight of all the wiper sets will just be equal to the weight of one wiper set, which is 0.5 pounds, times the number of wiper sets packed, which is x. So, we can substitute that into our equation as well. And in the end, we have y equals
We've come up with this wonderful equation, y equals 7 plus 0.5x. But my question for you is, what values am I allowed plug in for x in this equation? Strictly mathematically speaking, what is x allowed to equal here? Can it be any real number, any real number but 0, any real number but negative 3.5, only integer values or only whole numbers?
The answer is that x can equal any real number. We can pick any of the kinds of numbers that we've talked about, plug them in for x, and then get a value for y.
Think back to what x and y stood for when we originally developed this equation to help with Grant's suitcase problem. Considering that, does your answer for what it makes sense to plug in for x here change?
When we think about Grant's situation with his suitcase, x can really only equal a whole number. Remember that x stood for the number of wiper sets that Grant has packed in his suitcase. And since wiper sets are objects in the world, they're countable things. You can't really have half of a wiper set, or I guess you could, but Grant's only taking complete sets with him. Now, when I say that wiper sets can be counted, you might automatically think of the natural numbers. Since these are the counting numbers, 1, 2, 3, 4 and so on. But in principle, Grant could also pack zero glasses wipers. He could just not bring any wipers to the expo, probably wouldn't be that smart, but he could do it. And if we take this set of natural numbers and add zero to it, we get the whole numbers.
Now that we just said that x in the suitcase situation, can only equal whole numbers, how does that affect y? With that restricted set of values that x can equal, what will y end up equalling?
y will be greater than or equal to 7. We know that the smallest value x can take on will be zero, since that's the fewest number of glasses wipers Grant can put in his suitcase. And if he does pack that minimum number, and we plug zero in for x right here, this term will equal zero, and y will just equal 7. Every other number that we could plug in for x would be greater than 0, so y would be greater than 7.
Like we've done before, let's figure out some points to plot, so we can eventually graph this equation. Since we can fit a lot of glasses wipers in a suitcase, we're going to count them up by tens. I'd like you to find the corresponding weight of this suitcase or y value for each x value. I already done one of them for you just to make your life a little bit easier.
And our answers are 7 if x equals 0, 12 if x equals 10, 17 for x equals 20, 27 for x equals 40, and 32 if x equals 50.
With our filled out table, what do you notice about the y values as the x values increase? What numbers belong here according to the table?
Every time x increases by 10, y increases by 5. To get from 0 to 10, we have to add 10. And from 10 to 20, add 10 again, and so on and so forth for the rest of the x values. We're stepping by 10 every time. On the y side however, each number is 5 greater than the last one. So, the change from table entry to table entry, is 10 in the x column, but 5 in the y column.
Now that we've talked a bit about how our variables x and y are related in this equation we've developed for the total weight of the suitcase depending on the number of glasses wiper sets that Grant puts in it, I think that the next step to further understand the relationship here, is to graph our equation. Now, I know that I've done some graphing for the class already on the tablet. But since I really want to encourage you to graph on physical, actual graph paper, I'm going to just show you a demo of how I do that. Justmake sure that you've got the basics for setting up a graph and drawing a line totally down. So, I clearly has to switch pens since this one doesn't write on actual paper. Materials that are important for graphing include graphing paper, which is just paper with lines on it, that create a grid. A ruler, like this lovely one right here. And a pencil. I'm super picky about using pencils and not pens for graphing. I think it's really important so that you can make sure that you can fix any mistakes that you've made by erasing. The xy table we created earlier is going to be really useful for helping us graph points. But before we can graph those points, we need to set up our coordinate plane. So, we need to draw a set of axes, we can plot these points on. So, start out by lining the ruler up with one of the vertical lines on the sheet of paper. Over to the left-hand side, since all of the values that we need to graph in this graph are both positive in x and y. Sometimes, this won't be the case and you'll need to think about where to put your axes, and how far to draw the x-axis in both directions and the y-axis in both directions, so that you can fit all the important points on your graph. But for now, I have drawn a y-axis. And now, I line up the ruler horizontally, you can notice that it's positioned quite toward the bottom of the y-axis that I've drawn, since I don't really need any negative y values for this. And once I've got the ruler situated properly, in relation to this horizontal line r ight here, I can draw my x-axis. Wonderful, so now I have a y-axis and an x-axis. What's missing right now is some arrows. I know that the y-axis and the x-axis both extend infinitely in both directions, they both go from negative infinity to positive infinity. I also like to label the origin with a 0, and I label each axis as well with a variable it corresponds to. Great, now I have some axis. However, let's look at what numbers we actually need to plot. We need to plot from 0 to 50 in x and from 7 to 32 in y. We've got 0 laid out, that's great. But to go all the way to 50, I clearly can't have each one of these boxes stand for have quite a few boxes here, and I'm only graphing 6 points, I need to make every grid length on the x-axis stand for 5. And just for simplicity, I'll label every two grids with a 10. So, 10, 20, and so on, and so forth. I can do the same thing or a similar thing for the y-axis. Here, I only need to go up to 32 though, so I can probably space my numbers out a little bit more. I've labeled my y-axis that every 2 grid links stands for 5. And, of course, I've labeled it all the way up. Now, that we have our axes set up, we can actually start graphing points. This is sort of a tricky example to start with, so I'm sort of glad that I picked it. Because you'll notice that since there are numbers that end in 7 and in 2 on the y values, these aren't going to correspond exactly to grid lengths here. So, this is a great chance to see that sometimes when you're graphing, you don't have points that end up exactly where lines on the grid cross. You have to just be very careful and estimate as closely as you possibly can. To remind us of the coordinate or ordered pair notation that we'll often see when we're moving to graphs, I've written all the data in the table here in point notation. Okay, let's actually graph. First point, 0 7, so we start at 0 on the x-axis and move up 7 in y. The point halfway between 5 and 10 is 7 and a half, so we're going to be right below that. And I'll put a point right there. We have 10,12 so we go to 10 on the x, and 12, which will be just below this line right here, which is 12 and a half. And we continue just like this. Hm, something looks kind of funny to me. Let's check with a ruler to see if all of these points lie on a single straight line. Clearly, this one doesn't, so I must have done something wrong. It's really, really easy when you're graphing. Two, when you move over on the x-axis and then up to your y-coordinate, accidentally switch your hand position horizontally, and add to a graph in their own point. This is why we just need to check our work very carefully. Okay. So, that was the point 40 27,. So, I'll just graph that one again. 40, very carefully following this line up to the 27 mark, that looks better than the other point. So, you can see I must have just shifted my hand to right when I was graphing. Okay. Now, that I have all my points, let's make a line through them. Rulers are really helpful but you also have to make sure that you're using them as best you can. I like to put the point of my pencil on a point and then line the ruler up with the pencil tip, just make sure that the ruler's actually going to be in line with the points. I can do the same thing with the last point I'm going to graph. And now, I actually connect the points. Get a beautiful line. Wonderful. Now, since conceivably, Grant could pack more than 50 pairs of glasses wipers in his suitcase, I'm going to draw and arrow on the end of the line right here. This indicates that the graph could continue increasing in the x-direction and, in this case, the y-direction as well, past this last point that we graphed. I'm not going to put one over on this end though because for the purposes of the dealer we're dealing with, we're not going to have negative x values. Grant can't pack a negative number of glasses wipers in his bag. This would be very different though, if we were graphing other kind s of equations that don't necessarily correspond to things that can't have negative quantities. So, with most graphs, you'll see an arrow on the other end of the line as well. We'll see that in later examples.
Continuing in the vein of graphing equations, let's look at a different equation. y equals 3x minus 5. To move toward eventually graphing this, let's start by throwing out this xy table. I've already given you, lucky for you, all of the x values and for each of them, I'd like you to find the corresponding y value.
So here are our answers. The y values that go with these x values are negative
Thinking about this new table of values we created, can you complete this sentence? For the equation y equals 3 x minus 5, when x changes by some number, y changes by some other number. Fill in these two boxes.
Comparing each x value to the one that came before it, as we move down our table, every entry is one greater than the one that came before it. If we do the same thing to the y side of the table, we notice that every value is three greater than the one before it. So, when x changes by 1, positive 1, y changes by positive 3.
In order to set up our x and y axes to graph this equation right here, we need to think about how long we want each of them to be or what values you want them to extend through. Of course, we know that when we graph a set of x and y axes, we put arrows in the end to indicate that they actually extend infinitely long in each direction, but I am just interested in finding out what range of values for both x and y we should display on our graph. The goal here is to pick a range for x and possibly a different range for y, so that all of the values we have on our table are shown on our graph.
Since on our table, x varies from negative 3 to 3, we need to make sure that whatever we pick over here contains all those values. 0 to 10 and negative 10 to it's way bigger than this set, so I think the best choice is from negative 5 to only choice that works for y.
Finally, the time has come. We're going to graph y equals 3 x minus 5. I'm going to set up our x and y axis according to these ranges of values that you just picked. So, I've drawn our basic x and y cooridinates. That we are eventually, pretty soon, I promise, going to draw these points on. But remember, we need to label the crosshatches on either axis, so that we show what the scale of each axis is. So, I'm going to put a box here, a box here, box here, and a box here. These are tiny boxes. Yours will show up bigger on the screen, I promise. And I'd like you to tell me for these two boxes what the x-coordinate of each one is, and for these two boxes what the y-coordinate is for each. This is going to show, basically, how much moving from here to here on the y-axis counts as we're counting, and how much moving from here to here on the x-axis counts.
I get rid of those little boxes just to make our whole graph area look a little bit cleaner. Since our x values need to range from negative five to five on this portion of the axis that we've drawn, each of these grid lengths needs to count for one. So, when we step from zero, one step over, we need to label that one, two, three, and so on. And you see we get to five on the positive edge of the axis. Counting in the negative direction, we had the same scale. So, we have negative 1, negative 2. And you can see, I even made a mistake when I was drawing these axes. I made our x-axis one longer than it needed to be, which doesn't really hurt us, so I'm going to leave it as is. I apologize for my lack of perfection here. Now, in the y-direction, we need to be able to go from positive 10 all the way down to negative 20. We have 5 boxes up here and 10 boxes down here. So, that means we need to count by 2s with each step of teh graph. So, moving from 0 up one box, is going to actually be stepping 2 in the y-direction and then down into the same scale again. Now, again, the point of creating these scales and these limited links for either access is really just to make our lives easier a little bit when we're graphing. The x and y values for this graph do extend, actually, all the way from negative infinity to positive infinity. We're only going to graph part of that infinitely long line, in particular, the part that shows these values on this graph.
And now, finally, I would like you to actually graph this. Now, please first, just take a second, get out a piece of graph paper, create the same set of axis. I kind of hope you already did that before on your own anyway. And then, on that set of axis, graph the points that we have on the table. After you've got that done, connect them using a ruler with a line. After you've done that, come back to this quiz on the computer and select all of the boxes on this grid where you think that these points belong.
Now that you've had a chance to graph, it's my turn. Let's see if I can do this right. Because it works better for my brain, I've transferred all the data on the table to coordinate notation on the right-hand side of the table. That just helps me graph better, personally. So, I'm going to plot these points. We step over 3 in the negative direction on the x-axis, and down 14. Now, you want to make sure that the points, when you graph them on graph paper, as we saw in the previous example, are as accurately placed as possible. So luckily, for the scale that we've chosen for this graph and this equation, all of these points should lie directly on intersection points of vertical and horizontal lines on the graph paper. So, that's awesome. Negative 2, negative 11, and you can see I just made a mistake again. I forgot that we were counting by two's in the y direction, and I placed a point at 2,2 instead of 2,1. So, I've graphed a fair number of graphs in my life. So, if you make a mistake doing this, it's not a big deal at all. We just have to make sure we correct ourselves. And lastly, we have 3,4. Okay, great. These look to me like they're all in a pretty straight line, so let's get our ruler to make sure. I think we're safe. Let's connect them. And we have a gorgeous line that I will now draw an arrow on either end of. Remember, this arrow shows this line is extending forever out in either direction, continuing its pattern for all values of x. I've also drawn an arrow on either end of this line to show that it extends forever out in both directions.
Remember that one of the most powerful features of graphs is that they allow us to calculate x and y values for our equation that we may not have had previously. Of course, we can do that with the equation itself, but the graph lets us visualize this, not just calculate it. I'd like you to tell me, first off, what the y value of this equation is when x equals 4. And then second, when we have y equals 9 on our graph, what is x equal to?
Let's find where on our graph we have x equals negative 4. So I'll count over, negative 1, negative 2, negative 3, negative 4, on the x axis. And then slide down to the graph. What is that y coordinate? Maintain this vertical position, but sliding over to the y axis, we can see that this is halfway between negative means that this point right here on the graph is negative 4, negative 17. When Y equals negative 9, you can start on our Y axis and go all the way down to negative 9, and then slide over to the graph. Keep the same horizontal position of my pencil tip and slide up to the X axis. That is half way between negative 1 and negative 2, so that means that this point right here must be negative 1 and a half, negative 9. So when y is negative 9, x is equal to negative 1.5. Of course, you could also write this in fractional form if you wanted to. Of course, to make sure that our graph and the conclusions that we've drawn for it are totally accurate, we can always plug in these values to our equation, to figure out the value of the other variable.
Thinking about all of your recent graphing experiences, what is the minimum points that we need to draw a straight line? In addition to this, what is the minimum number of points we need if we want to draw a straight line but then also have one extra point to check that the line we drew is the line we actually wanted to draw?
To draw a straight line, we only need 2 points. We have point a and we have some other point, then the shortest path between them, is just a straight line. If we had just 1 point on its own, we wouldn't know what direction to draw the line in, unless we had another point that we knew we needed to aim toward. If we want to check that this actually the line that we were looking for, however, we need line. So, we need 3 points, if we want to check our original line. So, maybe this point right here, is the third point that I had in addition to these two. We can see if the original line I was starting to draw goes through this point as well. And voila, we have one continuous beautiful line. Let's say, for example, that I had 3 points, and I thought that they all lay on the same line, because I thought that I had done my xy table correctly. Unfortunately, when I bring my ruler into the picture, there is no way that I can get a line that I would draw along my ruler to contain all 3 of these points. So clearly, one of my coordinates must be messed up. So, a lot of the time, just to be safe, it's great to have 3 points that you think satisfy your equation, just to make sure your graph is correct.