Comparing linear and exponential relationships

0:00- [Instructor] We're told that Talia and Marcus take guitar lessons from different online platforms to learn new chords. Talia has already learned 20 chords on her own, and her guitar lessons increase her total by nine chords each week. Marcus has already learned two chords on his own, and his guitar lessons triple his total each week.

0:21According to each person, How many chords will they learn after three weeks? Pause the video, have a go at that before we do it together. Okay, so let's just think about this. I'll make a little table here actually. So let's look at, so this will be week, oh, maybe I should say after whichever week we're going to talk about.

0:47And then we have Talia and we have Marcus. So let's say at the end of week zero, end of week one, end of week two, end of week three, which is what they care about, learn after three weeks. So Talia has already learned 20 chords, so she's just going to start at 20.

1:05And we know that she increases by a total of nine chords each week. So after the first week, we'll add nine, 29. After the second week, we'll add nine more. That is 38. And after the third week, we'll add nine more, and that is 47. Now Marcus is starting at two chords, but then every week it triples his total.

1:31These are interesting guitar lessons. I'd like to take guitar lessons like that. So after the first week, we're gonna triple the total. So the total goes from two to six. So this is times three. And then we're gonna triple again at the end of the second week. So times three gets us to 18.

1:49And then we're gonna triple again at the end of the third week. So let's see, 18 x 3 is 54. So Talia, we're adding nine every time. Adding nine, adding nine. And Marcus, we're multiplying by three every time. So we've answered their first part of the question.

2:09Talia will have 47 total chords at the end of week three, and Marcus will have 54. So even though he started lower, he's now passing up Talia. And some of you might already appreciate that Talia is growing at a linear rate, adding or subtracting, in this case adding, the same amount after every equal time period, while Marcus is growing at an exponential rate.

2:30We're multiplying by a factor, by the same factor every time. Now there is a second part to this problem, so let's just take a look at that. Match each person to their graph. So you can pause this video, have a go at that before we do it together. Alright, there's a lot of different ways to look at this.

2:48You might say, "Who starts at 20? Who starts at two?" Well, Talia starts at 20, so that must be Talia's graph. So this must be Talia right over here. And then Marcus starts at two, so this must be Marcus, right over here, Graph B. Another way you could have thought about it is Talia is growing linearly.

3:07And this is linear. We can see after every week, after every week, we are increasing by nine. After every week we are increasing by nine, while Marcus is growing exponentially. We're growing by a factor of three after every week, two to six, six to 18, 18 to 54.