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Slope and intercept meaning from a table

Given a linear relationship in a table, Sal graphs the relationship to explore what the slope and intercepts mean in the given context.

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Video transcript

- [Instructor] We're told that Felipe feeds his dog the same amount every day from a large bag of dog food. Two weeks after initially opening the bag, he decided to start weighing how much food remained in the bag on a weekly basis. Here's some of his data. So we see after 14 days, there's 14 kilograms remaining. Then after another seven days pass by, so now we're at 21 days from the beginning, there's only 10.5 kilograms left. Then after 28 days there are seven kilograms left. All right, so we are going to try to use this data to start answering some interesting questions, and maybe we'll also try to visualize it with a graph. So the first thing that we might try to tackle is well how much food was in the bag to begin with if we assume that he's using the same amount of food every week. So pause this video and see if you can figure that out. How much food was in the bag to begin with if we assume that Felipe is feeding his dog the same amount every week? Okay now there's several ways to do this, but to help us visualize this, let me see if I can graph the data that we have and then see what would happen as we approach the beginning of this, of what's going on here, the dog feeding, and maybe as we go to the end as well. So let's see, this is my x-axis, this is my y-axis. I'm going to make x-axis measure the passage of the days, so number of days on the x-axis. And on the y-axis I'm going to measure, I'm gonna measure food remaining, and that is in kilograms. And let's see, it looks like maybe if my scale goes up to, let's make this five, 10, 15, 20, and then 25, I can make it a little higher, 25, I think this will be sufficient. And then we wanna go up to 28 days, and it looks like they're measuring everything on a weekly basis, so let's say that this is seven, 14, 21, and then 28. And they gave us some data points. So after 14 days, there's 14 kilograms remaining, so 14 days, there's 14 kilograms remaining, right over there. After 21 days, there's 10.5 kilograms remaining, 21 days, 10.5 is right about there. After 28 days there's seven kilograms remaining, so after 28 days, seven kilograms. And we're assuming the rate of the dog food usage is the same, that he's feeding his dog the same amount every week. And so this would describe a line, that the rate is going to be the slope of that line, and then if we can plot this line, if we know where that line intersects the x and y-axes, we might be able to figure out some other things. So actually let me draw a line here, let me see if I can use this little line tool to connect the dots in a reasonable way. So let's say it looks something like that, that's our line, that'll describe how quickly he is using his dog food. So let me make sure that this dot is, should be on the line as well. Now let's try to answer that first question, and think about how we might do it. How much food was in the bag to begin with? So what point here represents how much food was in the bag to begin with? Well that's the amount of food remaining at day zero, at the beginning of this, so that would be this point right over here, would describe how much food was in the bag to begin with. This would be the y-intercept, y-intercept is when our x value is equal to zero, what is our y value. And when we just look at it, the graph, it looks like it's a little bit over 20, but we could find the exact value by thinking about the slope, which is thinking about the rate at which the dog food is being depleted. We can see that every week, every week that goes by, or every seven days that goes by, it looks like we use 3.5 kilograms. Or another way to think about it, every two weeks it looks like we use an entire kilogram. So let me put it this way, when we go plus 14 days, plus 14 days, it looks like we use up, or the food remaining goes does by, goes down every two weeks it goes down by seven kilograms, seven kilograms, negative seven kilograms. So if we wanna figure out this exact value, we just have to reverse things. If we are going back 14 days, then we are going to go up seven kilograms. So if we were at 14, up seven kilograms, this right over here is the point zero comma 21. So how much food did Felipe start with in the bag? 21 kilograms, and we got that from the y-intercept. Now another question is how much is Felipe feeding his dog everyday. Pause this video and see if you can figure that out. Well we know every 14 days he's feeding the dog seven kilograms, so one way to think about it is, and we're really looking at the slope here to figure out the rate at which he's feeding his dog. So the slope is equal to our change in the y, so negative seven kilograms, every our change in the x, every 14 days, and so seven over 14 is the same thing as 1/2, so this is equal to negative 1/2 of a kilogram per day. So this tells us everyday the food remaining is going down half a kilogram, so that means he's feeding his dog, assuming his dog is eating the food and finishing it, that his dog is eating half a kilogram a day. And if we wanted to ask another question, how many days will the bag last? How would you think about that? And we know it's going to be out here someplace, if we just continue that line, because this point right over here, where our line intersects the x-axis, that would be our x-intercept, that is the x value when our y value is zero, and our y is the amount of food remaining. So we wanna know what day do we have no food remaining. And we could try to estimate it, or we could figure out it exactly. We know that every 14 days we use up seven kilograms. So if we are at seven, as we are right over here, and we go 14 days in the future, we should use up the remaining contents of the bag, so plus 14 days we're going to use up the remaining seven kilograms. And so this should happen 14 days after the 28th day, so this is going to be the 42nd day.