Modeling car value over time with exponential decay
About this video
A car bought for $24,000 loses 15% of its value each year, and the video reads the graph of this exponential decay. The y-intercept is the price when new, the graph is steep at first and flattens because 15% of a smaller value is fewer dollars, and the value approaches but never reaches $0 (a horizontal asymptote).
You'll be able to
- explain what the y-intercept means on a graph of value over time.
- explain why losing a fixed percent gives smaller dollar losses over time.
- describe a horizontal asymptote in context.
Key words
- exponential decay
- Decrease by the same percent (multiplying by the same factor) each time period.
- depreciation
- The loss in value of something, like a car, over time.
- y-intercept
- Where the graph crosses the vertical axis; the value at time zero.
- horizontal asymptote
- A horizontal line the graph gets closer and closer to but never reaches; here, a value of $0.
Transcript
- [Instructor] We're told a car is purchased new for $24,000. Each year, the car loses 15% of its value. And we can see that in this graph over here. Horizontal axis is time in years, vertical axis is value in dollars. And if the car is purchased new for $24,000, we should see that.
We should see that at time zero. We see indeed the graph is at $24,000. That's the .024, maybe I shouldn't use commas twice, that makes it confusing, 24,000. So the Y intercept represents what the value of that car is at time zero when it is new. And what you see here is actually exponential decay.
When we lose 15% every year, when the value of the car is high, 15% is a good bit. But as the value of the car gets lower and lower, 15% becomes smaller and smaller in absolute terms. And so when you go in, if you keep going forever, if you held this car forever and one could argue how long would a car actually last, we see that we are going to approach a zero value for this car.
We're gonna get closer and closer to this car being worthless for it to being worth zero. But we'll never quite approach that. And of course, this isn't exactly what happens in the real world, but this is a actual, not a bad model for how cars depreciate in the real world.
That initially, people often say, as soon as you take it off the lot, it depreciates. And that's not exactly being captured here, but a 15% annual decline is one way... It's a potential, a potentially realistic but simplified way of looking at a car's depreciation.
If you're interested, you can actually look up depreciation tables for different car models. And it's something important to look at, frankly, if you're buying, whether you're buying a new or a used car. But the key ideas here are, is we're looking at a graph.
And it's always important to think about, well, what does the graph really mean? Does it make sense? Time zero? Yes, that is how much the car cost new, $24,000. If I'm losing 15% every year, I'm not losing the same absolute dollars amount, it's 15% of wherever I am.
So initially I lose, my slope is steeper and it gets less and less and less steep. And over time, I'm approaching the car being worth zero, but I'm never quite getting there. That's our horizontal asymptote.
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