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### Course: Algebra 1 > Unit 12

Lesson 7: Exponential vs. linear models# Linear vs. exponential growth: from data

Sal constructs functions that model the growth of trees over time. To do that, he identifies which growth is linear and which is exponential.

## Want to join the conversation?

- This is a tough question to think about: Sal adjusted the exponential equation by making the exponent t/10 because the table represented increments of ten and not 1. But why didn't he just find the amount it would increase
*every individual*year like he found how much the linear equation did? The main idea is that both equations were "adjusted" in different ways to best model the problem, but there doesn't seem to be any rule about it.(28 votes)- He explained how to do it in the video after this (Linear vs Exponential growth: from data example 2).(0 votes)

- Aside from linear and exponential growths, are there other types of growth?(14 votes)
- Another user said "There are types of growth corresponding to every type of equation (e.g. quadratc growth, cubic growth, etc.), but these aren't often seen in nature.

Logistic growth is used frequently in biology. It models the population as the birth rate slowly decreases as the population approached a carrying capacity.On a graph, the curve of logistic growth is s-shaped."(4 votes)

- In all the videos, Sal showed that an exponential function is of the form f(x) = a * b^x, but then here, all of a sudden, b is raised to a fractional power. Can we do that? Also, out of curiosity, how could we model that so that the exponent is not fractional?(6 votes)
- In exponential equations, the exponent can be any real number (including fraction and decimals). Depending on the base value and the fraction, you may be able to figure it out without a calculator like: 8^(1/3) = cube root(8) = 2. In other situations, the exponent will create an irrational number and you will need to approximate its value using a scientific calculator.

Hope this helps.(3 votes)

- I really wanted to ask about the multiplication sign in B(t)=8·4 t/10, I'm not really sure what that means because I think it could be a plus?

Please clear up for me. Thx!(3 votes)- Sal is correct, and you have one slight problem in your equation, it should be B(t) = 8 * 4^(t/10), so it forms an exponential with 8 as the initial value, 4 as the base (Multiplier) and exponent of t/10.(8 votes)

- Aside from linear and exponential growths, are there other types of growth?(3 votes)
- There are types of growth corresponding to every type of equation (e.g. quadratc growth, cubic growth, etc.), but these aren't often seen in nature.

Logistic growth is used frequently in biology. It models the population as the birth rate slowly decreases as the population approached a carrying capacity.On a graph, the curve of logistic growth is s-shaped.(5 votes)

- A found another way to model the last table! Here it is:

f(t)=2^3+2t/10 where "t" is the number of years passed!:)(5 votes) - Everything was making sense to me until2:27in the video. I begin to get lost when Sal puts 34 + 4. Why did he do that? Is he just saying 4 is the constant? and 3t / 12 is 4t?(3 votes)
- 4 is the rate of change. there are approximately 12 more branches every three years, so the addition each year would be 4, because 12 branches divided by three years is 4 branches per year. hopefully that makes sense.(3 votes)

- at2:30, where did you get 4 from for the oak tree?(2 votes)
- The number of branches increased by 12 every 3 years.

So, over 1 year, it increased by 4 branches.(5 votes)

- Why do we put t over 10 instead of just t?(2 votes)
- Because the number of branches is multiplied by 4 every
**10**years. Simply putting t would result in a function that multiplies by 4 every year.(3 votes)

- couldn't it just be 0.4^t instead of 4^t/10(0 votes)
- No, (0.4)^t is not equivalent to 4^(t/10).

For example, if t = 10, then (0.4)^t is extremely close to zero, but 4^(t/10) is 4.

Also, the values in the table are growing (increasing) with time. Note that (0.4)^t would represent exponential decay, instead of growth, since 0.4 is less than 1.

Have a blessed, wonderful day!(10 votes)

## Video transcript

- [Voiceover] The number
of branches of an oak tree and a birch tree since
1950 are represented by the following tables. So for the oak tree we
see when time equals zero has 34 branches. After three years it has 46 branches, so on and so forth. And in the birch tree
they give us similar data. At the beginning has eight branches. In 10 years has 33 branches
and they give us all of that. What I want to think about in this video is how should we model these? If we want to model these with functions and the choices we'll give ourselves, there are other options, but the choices we'll give
ourselves in this video are linear, and linear versus exponential functions. Which of these are going to be better for modeling this data? So let's first look at the oak tree. A key to realization is,
whenever I have a fixed increase in time, so each of these steps, this is plus three years. So it's a fixed increase in time. What happens to my number of branches? Is it going to be a fixed change, or roughly a fixed change in which case a linear model might be good, or is it going to be a
change that's dependent on where we were? So what am I talking about? So 34 to 46, that is plus 12. 46 to 59 is plus 13. 59 to 70 is plus 11. 70 to 82 is plus 12. So at first you mention,
well this isn't exactly a fixed change. These numbers they seem to
average right around 12, but when you're looking at real world data you're never gonna get
something that is exact. The models are just going
to give us a good fit. Are going to give us a good approximation of the behavior of the
number of branches over time. For me, this is pretty close
to a constant 12 branches a year. So I would construct a linear model here. I would say here branches as a function of time. Let me be clear, this
isn't 12 branches per year, this is 12 branches every three years. This is 13 branches over three years. This was 11 branches over three years. We're going to average 12
branches over three years. So the number of branches we have, we're gonna start at 34 branches and then minus. If we have 12 branches every three years, that's four branches every, or I should say plus. Four branches every year. And you could test this out. B of zero is gonna give us 34 branches. B of 12. Let's just really test out
the extreme part of the model. B of 12 is going to be 34 plus 48, which is equal to 82. So this model works quite well. It's gonna have a couple of places where it's not exactly fitting the data, but it fits it quite, quite well. This is the linear model. So this one is linear. We also got the birch tree. So time equals zero, so
fixed change in time. Alright, so we have a fixed change in time every time we are moving
into the future, a decade. Let's see, our change in branches. We go from eight to 33. So what is that? That is plus 25 branches. Then we go from 33 to 128. Well that's way more than 25 branches. That's going to be what? Five less than 100. So that's gonna be plus 95 branches. So this clearly is not a linear model. And so let's think in terms
of an exponential model. How much do we have to
multiply to go from... Did I do that right? 128 minus, yeah, if it was 133 then it would be 100 and it's five less than that, okay. So now let's think about it in terms of an exponential model. In terms of an exponential model what do we have to multiply for each step? So if we have a constant step in time, what do we multiply for how much we increase our branches? So if we go to eight to 33, it's gonna be approximately four. It's gonna be a little bit more than four. 33 to 128, well that's gonna be a
little bit less than four, but it's approximately four. 33 times four would be
132, so we're close. 128 to 512, that's exactly four, right? That's exactly 120 times four is 480 plus 32, yup, that is exactly four. So times four. And so it looks like we
keep multiplying by four every 10 years that go by. So one way to think about it is we could say here, BFT, the branches of T, our initial condition,
our initial state is eight and now we could say our
common factor is four. But if we want T to be in years, well every 10 years we
multiply by a factor of four. So T has to go to 10 before we increase the exponent to one, or has to go to 20 until
this exponent becomes two. So eight times four to the T over 10 power seems like a pretty good model. And you could even verify
this for yourself if you like. Try out what B of 30 is going to be. B of 30 would be eight times four. 30 divided by 10 to the third power and what is that going to be? That's going to be four
to the third is 64. 64. Eight times 64 is, it's 480 plus 32. It is 512. So once again, this exponential model for this data does a pretty good job.