Comparing scale in proportional relationship graphs
About this video
A parking garage charges $8 per hour, and Theo and Bianca graph the cost using different axis scales. Theo claims Bianca's flatter-looking line shows a cheaper garage, but the video shows both graphs have the same slope, $8 per hour, because the scale of the axes changes how a line looks, not the rate it shows.
You'll be able to
- explain that the slope of a cost graph represents the price per unit (here $8 per hour).
- calculate slope from a graph using change in y over change in x.
- explain why different axis scales can make the same relationship look more or less steep.
Key words
- slope
- Change in y divided by change in x; here it is the cost per hour.
- scale
- How much each step or hash mark on an axis stands for.
- rate
- How much one quantity changes for each one unit of another, like $8 per hour.
Transcript
- [Narrator] A parking garage charges $8 per hour. Theo and Bianca each create a graph to show the cost y for x hours of parking. Theo creates axes that go from zero to six on the x-axis and zero to 48 on the y-axis. Bianca creates axes that go from zero to 30 on the x-axis and zero to 240 on the y-axis.
All right, so they're plotting the same thing. They're just doing different scales for the axes. Theo says Bianca's graph shows a cheaper parking garage because her line looks less steep. Is Theo correct? So pause this video and think about that. All right, so I actually will show you what those graphs will look like, but before we even look at those graphs, you shouldn't, based on how you graph something, it shouldn't show one being cheaper than the other.
Assuming they graphed it correctly, we should both see $8 per hour as essentially the slope of the graphs. For every increase one hour, we should see an increase in $8 of cost. So assuming they graphed them correctly, you should not see one of the graphs being cheaper than the other.
Now, depending on how they graphed it and how much they increased either in the vertical or the horizontal axis for each hash mark or how they labeled it, yes, it's possible that one of them looks steeper than the other, but they should both show a slope of $8 per hour.
So I'll say that Theo is not correct and they should both show the same parking cost of $8 per hour. So now let's look at the actual graphs and feel good about that. So here's Theo's graph. He goes from zero to six hours on the horizontal axis and it goes from zero to 48.
And if you look at the cost, you can see every hour, we have an incremental increase in $8. Every hour, an incremental increase in $8. The slope here is $8 per hour. Now look at Bianca's graph. And actually, these look pretty much the same even when you optically look at them, they look like they have the same slope, but let's actually just verify that.
So when we increase by five hours, our cost increases by eight, sorry, by 40. But if you take 40 divided by five, our change in y over change in x, that's still eight. Once again, you increase by another five, you increase by another 40. 40 over five is still eight.
Both of these have a slope of eight because they're both graphing the same idea that the parking garage charges $8 per hour.
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