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### Course: Algebra 1 > Unit 12

Lesson 1: Exponential vs. linear growth# Exponential vs. linear models: verbal

Linear growth occurs at a constant rate, with equal increments added or subtracted over time, while exponential growth involves a constant multiplier that drives an increase or decrease over time. We can look at the type of change over time to see if given example represents linear or exponential growth. Created by Sal Khan.

## Want to join the conversation?

- How would you write the problem that states "The number of wild dogs in Arkansas increases by a factor of 3 every 5 years" as an equation?(10 votes)
- You can't write the full equation, since you need the initial value, but we can just call that
`a`

. Then you need the ratio, which is 3, so using our`a•rˣ`

form it seems like the equation should be`N = a•3ˣ`

. However, we probably want to have it as a function of the number of years, but right now`x`

is the number of "5 year periods". Since the number of 5 year periods is just the number of years divided by 5, we can replace`x`

with`y/5`

, where`y`

is the number of years, so we have`N = a•3^(y/5)`

. This is a fine form to leave the answer in, but if we want to write this in the pure exponential form, we can split the`y/5`

exponent into a`1/5`

and a`y`

and so:`N = a•(3^⅕)^y = a•(⁵√3)ʸ`

.(27 votes)

- At0:42, how did Sal know to increase the function by 1.05?(10 votes)
- The original weight increases by 5%. 5% = 0.05, but we are adding that to the original weight, so 1+0.05 = 1.05.

If the original weight had been 100 and we increased it by 5%, we would be increasing it by 5% of 100, which is 5% * 100 = 5, so we add 100 + 5% * 100 = 1* 100 + 0.05 * 100 = (1+0.05) * 100 = 1.05 * 100 = 105.(10 votes)

- I'm not sure why dividing one by a fraction makes the fraction negative and I'm having a hard time understanding. If for example, you divide one by one-half wouldn't that be two not negative one half? 1/ 1/2 = 2?(9 votes)
- Where did you see that 1 divided by a fraction makes the fraction negative? It doesn't. Can you tell me where you saw this?

Yes, 1 / (1/2) = 2. This is correct.(6 votes)

- Why is it worth comparing these two types of functions? Are linear and exponential models more important than other functions?(2 votes)
- As far as in algebra and further, you will be dealing with a lot of graphs and functions. In fact, Calculus is ALL about graphs and functions. Thus it is important for you and everybody to build up the basics of graphs and get acquaintance of how the look and what the difference is. Being able to distinguish the type of a graph just by reading is an enormous profit for you both at school and outside school!

To your second question, linear and exponential graphs does not necessarily have to be "most important". It is only that it is simply and easy to understand so you get the basis for the basics.(15 votes)

- is the equation y=40(1.05)^i here i is the week, correct?(6 votes)
- Yes, that would be the equation for the 1st problem in the video.(6 votes)

- not really a question. I really like how the equation/ formula is explained in great detail.(8 votes)
- So linear functions increase by constant VALUES, while exponential functions increase by constant PERCENTAGES?(4 votes)
- Linear equations increase by a constant slope, but exponential equations increase by a constant exponent or power. For example, y = 2x + 1. It starts from 1 and each x is multiplied by 2. On the other hand, exponential equations of form y = x^2 increase each x by the power of 2.(4 votes)

- Can a Linear Function be expressed into an Exponential Function? In other words, Can you convert it into an Exponential Function? Thanks a lot.(3 votes)
- For a linear function, e.g. "y = k*x", as x increases, y increases exactly k times as much. So y always increases at exactly the same rate at all values of x.

For a quadratic function, e.g. "y = x^2", at each value of x, y is increasing at a rate of 2*x (from calculus). Whatever x is - "-6", "+2 million", whatever, y is changing at a rate twice that as x changes.

Because of this stark difference, no quadratic function can be expressed as any linear function, nor can any linear function be expressed as any quadratic.

For similar reasons, the same thing is true of other degree polynomials, to either integral or real degrees (unless the degree is 0, where its a constant function (a special case of linear); or 1, where it's linear).

It seems to me that polynomials of any degree would be incompatible in the same way with polynomials of other degrees - but I don't know any details about this. Maybe someone else can clarify.

Now we come to exponential functions. In polynomials the powers are constants and the independent variable x is the base, which is allowed to vary. In exponentials, the base is any positive constant not = 1, and the power is the variable x (any real number), or a function of x. So as x increases, a^x is raised to higher and higher powers of a.

To compare, say, 2^x and x^2; in x^2, as x increases to x+1, y increases to x^2+2x+1. whereas 2^x, doubles to 2^(x+1) = 2*2^x. Since as x gets large x^2 is much larger than 2x+1, It can be seen that the increase of x^2 is insignificant compared to the doubling of 2^x as x gets large. So since 2^x is increasing at much faster rates it must overtake x^2 at some point, and thereafter renders it insignificant. I'm going to guess that any exponential will overtake any polynomial, regardless of degree, at some value of x.

If so, then exponentials can't be expressed in terms of any polynomial, linear or otherwise, and vice versa.

You could convert by raising a base to a polynomial power, or taking a log of an exponential whose power is a polynomial function of x.(5 votes)

- So can we rule that multiplication/division is always exponential and addition/subtraction is always linear?(2 votes)
- Well, you have the right idea, but you are not asking the question quite correctly. A linear function such as y = 4x + 5 in fact has a multiplication component (4*x) and an addition component ( + 5). But multiplication is a short cut to addition (and relating linear to arithmetic sequences), to get from one value to the next, you have to add 4 to previous term (same for subtracting with y = - 4x + 5). Similarly, exponents are a short cut for multiplication such as y = 2^x + 5 (and relating exponential to geometric sequences) to get from one value to the next, you have to multiply a number of 2s together and then add 5 (same for division to have 2^-x + 5).(6 votes)

- how is that different from the geometric sequence? how would it be worded if it was a geometric sequence?(2 votes)
- They are very similar. Both involve terms increasing by some factor.

The difference lies in the fact that we call one an exponential**function**and the other a geometric**sequence**. (Problems will probably specify which one they want you to use).

A function is something that takes an input and creates an output. You would write an exponential function like this: f(x)=2^x.

A sequence, on the other hand, is an ordered list of numbers that is defined by some relationship. Unlike functions, sequences can be expressed explicitly or recursively. An example of a geometric sequence is 1, 2, 4, 8, 16, 32.

Another difference between the two is that exponential functions are continuous. The graph is a smooth curve, and you can have inputs like 0.5. For sequences, though, n can only equal whole numbers.

The differences are similar for linear functions and arithmetic sequences too.

Hope this helps!(6 votes)

## Video transcript

Voiceover:A newborn calf
weighs 40 kilograms. Each week its weight increases by 5%. Let W be the weight in
kilograms of the calf after t weeks. Is W a linear function or
an exponential function? So, if W were a linear
function, that means that every week that goes by, the weight would increase by the same amount. So let's say that every week that went by, the weight increases ... Or, really, they're
talking about mass here. The mass increased by 5 kilograms. Then we'd be dealing
with a linear function. But they're not saying that the weight increases by 5 kilograms. They're saying by 5%. After one week it'll be
1.05 times 40 kilograms. After another week it'll
be 1.05 times that, it'll be 5 percent more. After the next week
it'll be 1.05 times that. So really, if we really
think about this function, it's going to be 40 kilograms times 1.05 to the t power. We're compounding by 5% every time. We're increasing by a factor of 1.05. Or another way of thinking
about it, by a factor of 105% every week. Because we have that growth by a factor, not just by a constant
number, that tells us that this is going to be
an exponential function. So let's see which if these
choices describe that. This function is linear, no, we don't have to even read that. This function is linear, nope. This function is exponential
because W increases by a factor of 5 each
time t increases by 1. No, that's not right. We're increasing by 5%. Increasing by 5% means
you're 1.05 times as big as you were before increasing. So it's really this
function is exponential because W increases by a factor of 1.05 each time t increases by 1. That, right over there,
is the right answer. Let's try 1 more of these. Determine whether the quantity described is changing in a linear fashion or an exponential fashion. Fidel has a rare coin worth $550. Each year the coin's
value increases by 10%. Well, this is just like the last example we saw. We're increasing every year that goes by as we increase by a factor of 1.1. If we grow by 10%, that's
increasing by a factor of 110% or 1.1. So this is definitely exponential. If it was increasing $10 per year, then it would be linear. But here we're increasing by a percentage. Your uncle bought a car
for 130,000 Mexican pesos. Each year the value of the car decreases by 10,000 pesos. So here we're not multiplying by a factor, we're decreasing by a fixed amount. 1 year goes by, we're at 120,000. 2 years goes by we're at 110,000. So this is definitely a linear ... This can be described by a linear model. The number of wild hogs
in Arkansas increases by a factor of 3 every 5 years. So a factor of 3 every 5 years. They're not saying it increases by 3 hogs every 5 years. We're multiplying by 3 every 5 years. So this is definitely ... This one right over here
is going to be exponential. Then, finally, you work as
a waiter at a restaurant. You earn $50 in tips every day you work. Well, this is super ... This should jump out as very linear. Every day you work, another $50. Work 1 day, $50. 2 days, $100. So forth and so on. They're not saying that
you earned 50 times as much as the day before. They're not saying that
you earned 50% more. They're saying that you're increasing by a fixed quantity. So this is going to be a linear model.