Slope, x-intercept, y-intercept meaning in context | Algebra I | Khan Academy

We're told Glenn drained the water from his baby's bathtub. The graph below shows the relationship between the amount of water left in the tub in liters and how much time had passed in minutes since Glenn started draining the tub. And then they ask us a few questions: How much water was in the tub when Glenn started draining? How much water drains every minute? Every two minutes? How long does it take for the tub to drain completely?

Pause this video and see if you can answer any or all of these questions based on this graph right over here.

All right, now let's do it together. Let's start with this first question: How much water was in the tub when Glenn started draining? So what we see here is when we're talking about when Glenn started draining, that would be at time t equals zero. So time t equals zero is right over here. And then, so how much water is in the tub? It's right over there.

And this point, when you're looking at a graph, often has a special label. If you view this as the y-axis, the vertical axis is the y-axis, and the horizontal axis is the x-axis. Although when you're measuring time, sometimes people will call it the t-axis, but for the sake of this video, let's call this the x-axis. This point at which you intersect the y-axis tells you what is y when x is zero, or what is the water in the tub when time is zero.

So this tells you the y-intercept here, tells you how much, in this case, how much water we started off with in the tub. And we can see it's 15 liters, if I'm reading that graph correctly.

How much water drains every minute? Every two minutes? Pause this video. How would you think about that? All right, so they're really asking about a rate. What's the rate at which water's draining every minute?

So let's see if we can find two points on this graph that look pretty clear. So right over there at time one minute, looks like there's 12 liters in the tub. Then at time two minutes, there's nine liters.

So it looks like as one minute passes, we go plus one minute, plus one minute. What happens to the water in the tub? Well, it looks like the water in the tub goes down by—from 12 liters to 9 liters—so negative 3 liters. And this is a line, so that should keep happening.

So if we forward another plus one minute, we should go down another three liters, and that is exactly what is happening. So it looks like the tub is draining three liters per minute. So draining, draining three liters per minute. And so if they say every two minutes—well, if you're doing three liters for every one minute, then you're going to do twice as much every two minutes. So six liters every two minutes.

But all of this, the second question, we were able to answer by looking at the slope. So in this context, the y-intercept helps us figure out where we started off. The slope is telling us the rate at which the water—in this case—is changing.

And then they ask us, how long does it take for the tub to drain completely? Pause this video and see if you can answer that.

Well, the situation in which the tub has drained completely means that there's no water left in the tub. So that means that our y-value, our water value, is down at zero.

And that happens on the graph right over there. And this point where the graph intersects the x-axis, that's known as the x-intercept. In this context, it says, hey, at what x-value do we not have any of the y-value left? The water has run out.

And we see that happens at an x-value of five. And but that's giving us the time in minutes. So that happens at five minutes. After five minutes, all of the water is drained. And that makes a lot of sense: if you have 15 liters and you're draining three liters every minute, it makes sense that it takes five minutes to drain all 15 liters.