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### Course: Algebra (all content)>Unit 11

Lesson 21: Distinguishing between linear and exponential growth (Algebra 2 level)

# 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.
• He explained how to do it in the video after this (Linear vs Exponential growth: from data example 2).
• Aside from linear and exponential growths, are there other types of growth?
• 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."
• 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!
• 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.
• 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?
• 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.
• Aside from linear and exponential growths, are there other types of growth?
• 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.
• 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!:)
• Everything was making sense to me until in 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?
• 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.
• at , where did you get 4 from for the oak tree?
• The number of branches increased by 12 every 3 years.
So, over 1 year, it increased by 4 branches.
• couldn't it just be 0.4^t instead of 4^t/10
• 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!