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### Course: Algebra 1 > Unit 12

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

Determining whether real world model is linear or exponential, where the model is given as a table.

## Want to join the conversation?

- Is there a quicker method to find whether something is exponential or linear, or is this the quickest way?(12 votes)
- On a graph, an exponential function is always curved (like a gradual slope) and a linear function is always straight (like a line). Hope that helps! :)(38 votes)

- What if the x-values increase at random? A sequence like 1, 5, 6, 9, 10, 17 et cetera.(10 votes)
- You just need to find the slope using two points. You just need to find the slope between x=1 and x=5, x=5 and x=6, x=6, x=9, etc. If the slopes are the same, it's linear and if they're different, it's exponential.

Hope this helps! If you have any questions or need help, please ask! :)(13 votes)

- Are there any quicker ways(7 votes)
- Given only a table of numbers, there is not a quicker way. If you have a graph, it is easy to see that exponential graphs have a curve whereas linear graphs are completely straight.(7 votes)

- At about2:52minutes into the video, Sal said; "It seems like we're multiplying by a slightly lower factor".

What does he mean by this?

How can he tell by just looking at the table, that it's a slightly lower factor?(4 votes)- He is explaining why it isn't an exponential function by showing that the factors between each number isn't constant.

He didn't just find it by looking at the table, he probably used a calculator off-screen while not recording this video.(5 votes)

- By approximating the cost differences to 7 (in thousands of dollars), is the function
*mathematically*a linear function?(4 votes)- I don't know if I'm right, but here's just my thoughts:

If when you're plotting the points on the graph and you just increase by 7 (you've already rounded it), then yes, the function will be linear.

If you plot the points precisely on the graph, it would look very much like a linear function, except the slope between each points would vary by a little bit. But to me, I'd just say it's a linear. But I think you can also call it a piecewise graph.(4 votes)

- Can an exponential function be dividing?(4 votes)
- @khana3105

Good Question! Explanation: These exponents have the same base, x, so they can be divided. To divide them, you take the exponent value in the numerator (the top exponent) and subtract the exponent value of the denominator (the bottom exponent).

Also, The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents.

Here's some rules to help you out.

1. ax ay = a. x+y

2. ax/ay = a. x-y

3. (ax)y = a. xy

4. axbx=(ab) x

5. (a/b)x= ax/b. x

6. a0=1.

7. a-x= 1/ a. x

Hope this helps.(3 votes)

- How do you valuate when an exponential function and a linear function are equal?

I have only seen examples with tables, but isn't there a way to equal the two functions and isolate the variable?(4 votes)- Yes if you look at the chart and try to find the starting number(4 votes)

- are there any slower ways to do this?(2 votes)
- Isnt there a quicker method or formula for this problem?(2 votes)
- Horses,

Graphing might be quicker, but you would have to evaluate a few points in order to draw the graph.(1 vote)

- What if it decreases by an uneven number?(1 vote)
- Then your function isn't linear. It may be another form of function(3 votes)

## Video transcript

- [Instructor] The table
represents the cost of buying a small piece of land in a remote village since the year 1990. Which kind of function best
models this relationship? And I'm using, this is an example from
the Khan Academy exercises, and we're really trying to pick between whether a linear model or a linear function
models this relationship or an exponential model or exponential function will model this relationship. So like always, pause this video, and see if you can figure
it out on your own. All right, so now let's
think about this together. So as the time goes by around this, the time variable right over here, we see that we keep
incrementing it by two. Go from zero to two, two to four, four to six, so on and so forth. It keeps going up by two. So if this is a linear relationship, given that our change in time is constant, our change in cost should
increase by a constant amount. Doesn't have to be this constant, but it has to be a constant amount. If we were dealing with an
exponential relationship, we would multiply by the same amount for a constant change in time. Let's see what's going on here. Let's just first look at the difference between these numbers. To go from 30 to 36.9, you would have to add 6.9. And to go from 36.9 to 44.1, what do you have to add? You have to add 7.2. And now to go from 44.1 to 51.1, you would have to add seven. Now to go to 51.1 to 57.9, you are adding 6.8, 6.8. And then finally, going from 57.9 to 65.1, let's see, this is almost eight, 7.1, this is what, 7.2 we're adding, plus 7.2. So you might say, "Hey, wait, "we're not adding the exact
same amount every time." But remember, this is intended to be data that you might actually get
from a real-world situation. And the data that you get
from a real-world situation will never be exactly a linear model or exactly an exponential model. But every time we add two years, it does look like we're
getting pretty close to adding 7,000 dollars in cost. 6.9 is pretty close to seven. That's pretty close to seven. That is seven. This is pretty close to seven. That's pretty close to seven. So this is looking like
a linear model to me. You could test whether
it's an exponential model. You see, well, what factor
am I multiplying each time? But that doesn't seem to be as, this doesn't seem to be
growing exponentially. It doesn't seem like we're multiplying by the same factor every time. It seems like we're multiplying
by a slightly lower factor, as we get to higher cost. So the linear model seems
to be a pretty good thing. If I see every time I
increase by two years, I'm increasing cost by
6.9 or 7.2 or seven. It's pretty close to seven. So it's not exactly the cost, but the model predicts it pretty well. If you were to plot these on a, on a coordinate plane and try to connect the dots, it would look pretty close to a line, or you could draw a line
that gets pretty close to those dots.