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### Course: Algebra 1>Unit 12

Lesson 1: Exponential vs. linear growth

# Exponential vs. linear growth

For constant increments in x, a linear growth would increase by a constant difference, and an exponential growth would increase by a constant ratio.

## Want to join the conversation?

• What's the difference between geometric sequences and exponential functions?
• A sequence would be like a bunch of dots on a graph at all of the natural numbers for x, the exponential function related to that sequence would be like connecting the dots and going back into the negative numbers also. They are related, but not the same.
• When Sal is giving the Exponential Function example, I noticed when he was saying that it increased by 2, then 6, then 18, you could also see that that 2*3=6, 6*3=18, and so on. Is this another way to find out if a given table is exponential or not, or does this work for only a few types of tables?
• The difference between the terms is called a common ratio. In your case, the common ratio is 3, because every time you get from `f(n)` to `f(n+1)` (to get to the next term), you multiply by 3. For instance, 2 times 3 is 6, and 6 times 3 is 18.

So given any table, to check whether the relation is exponential, just divide each term by the one before it. Say your sequence is 5/3, 5, 15, 45, 135, ... Is it exponential?

Here's the table for the sequence:
`` n  | 1 | 2 | 3 | 4 | 5 ----|---|---|---|---|---a(n)|5/3| 5 | 15| 45|135``

Now we divide consecutive terms. 5 divided by 5/3? 3. 15 divided by 5? Also 3. Repeat for every term in the series. If each term is multiplied by the same number (remember, it's called a common ratio) to get to get to the next, we know that the relation is exponential.
• I'm confused. Isn't Exponential Function as same as Geometric Sequence?
• An exponential function is a function where a fixed number is raised to every x. In other words, you pick a number, and each x on the axis is the power that the number is raised to in order to get y. A geometric sequence is a sequence where every x is multiplied by the same, fixed number. f(n^x) is exponential, f(nx) is geometric.
• x 15 16 17 18
y 10 20 40 70
Is the relationship linear, exponential, or neither?

That is a KA test question. Would anyone mind explaining how it is neither exponential or linear? Obviously not linear but how is it not exponential? Would this not look like a parabola if graphed? If it's not exponential, what is it? In a previous vid, Sal used an example and said that the values don't have to be exact and it can still be one or the other.
• Note, first of all, that the x values increase by the same amount each time. However, the y values neither always increase by the same amount nor always grow by the same factor (for example, 40-20 is not equal to 20-10, and also 70/40 is not equal to 20/10). So the relationship is neither linear nor exponential.

The relationship could be quadratic (parabola) because, while the differences between consecutive x values are constant, the differences between consecutive y values (10, 20, 30) are increasing at a constant rate.
• How is the exponential relationship not a different version of a linear relationship?
• A linear relationship has a constant rate of change. If you take any 2 points on a line, the slope found would be the same if you picked a different 2 points from the line. The graph is also a straight line.

An exponential relations grows / reduces on an accelerated basis. The graph will be a curve, not a straight line. You can find the average rate of change (the slope between 2 points), but it would be different from the slope found between another 2 points on the curve.
• what's the difference between an Exponential function and a geometric sequence?
• The equations look the same. The difference is in the acceptable input values and their graphs.

An exponential function accepts any real number as an input value. It's graph is a smooth continuous exponential curve (so no gaps).

A geometric sequence accepts natural numbers as its input values because the input is which term you are finding. If you graph the points created for the geometric sequence, they would fall on the corresponding exponential graph, but there are gaps in between the points.

Hope this helps.
• Would I be correct in saying that an example of neither Exponential nor Linear relationships would be if it does not increase at any constant/ fixed ratio?
• Exactly right!
(1 vote)
• Is there any difference between exponential and linear growth? Is there any difference between logarithmic growth? I would assume that logarithmic growth is similar to exponential growth.
(1 vote)
• Linear growth is constant. Exponential growth is proportional to the current value that is growing, so the larger the value is, the faster it grows. Logarithmic growth is the opposite of exponential growth, it grows slower the larger the number is.
• Linear growth is a fundamental concept that forms the basis of many mathematical and scientific principles. It refers to a steady, constant increase in a quantity over time, where the rate of increase remains the same. One of the most common examples of linear growth is the increasing size of a population. As the number of individuals in a population grows, the rate of increase is consistent, resulting in a linear growth pattern.

Another example of linear growth can be seen in the world of finance. When an investment grows at a fixed rate over time, it is said to be growing linearly. This type of growth is important in financial planning and forecasting, as it allows investors to predict future returns with a high degree of accuracy.

Linear growth can also be observed in scientific phenomena. For example, the growth of a crystal can follow a linear pattern, where the crystal grows at a consistent rate over time. Understanding the principles of linear growth can help scientists predict the behavior of these phenomena and develop new theories and models.
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Exponential growth can be observed in a variety of contexts, from population growth to compound interest. In algebra, exponential growth is often modeled using the function f(x) = a^x, where a is a positive constant that determines the rate of growth. As x increases, f(x) grows at an increasingly rapid rate, leading to explosive growth over time.

One of the key features of exponential growth is that it is self-reinforcing. As the function grows, the rate of growth itself increases, leading to even faster growth in the future. This can lead to some surprising and counterintuitive results. For example, if you start with a penny and double the amount every day, you would have over \$10 million after just 30 days!

Exponential growth is also closely related to logarithmic functions, which are used to model exponential decay. In fact, the logarithmic function is the inverse of the exponential function, meaning that it "undoes" the effects of exponential growth. Logarithmic functions are often used in finance and economics to model the decay of assets over time, as well as in chemistry to model radioactive decay.
• Is a linear relationship the same as an arithmetic sequence or a geometric sequence?
• An arithmetic sequence is a linear relationship and thus will be a linear function, but sequences are discrete, not continuous. There are other linear relationships that are not arithmetic sequences such as a linear equation (y = mx + b), so all arithmetic sequences are linear relationships, but not all linear relationships are arithmetic sequences.

## Video transcript

- [Instructor] So I have two different xy relationships being described here. And what I would like to do in this video is figure out whether each of these relationships, whether they are either linear relationships, exponential relationships, or neither. And like always, pause this video and see if you can figure it out yourself. So let's look at this first relationship right over here. And the key way to tell whether we're dealing with a linear, or exponential, or neither relationship, is think about, okay, for given change in x, and you see, each time here, we are increasing x by the same amount. So we're increasing x by three. So given that we're increasing x by a constant amount, by three each time, does y increase by a constant amount? In which case, we would be dealing with a linear relationship. Or is there a constant ratio between successive terms when you increase x by a constant amount. In which case, we would be dealing with an exponential relationship. So let's see. Here we're going from negative two to five. So we are adding seven. When x increases by three, y increases by seven. When x is increasing by three, y increases by seven again. When x increases by three, y increases by seven again. So here, it is clearly a linear relationship. Linear relationship. In fact, you can even, relationship, you could even plot this on a line if you assume that these are samples on a line. You could think even about the slope of that line. For a change in x, for a given change in x, the change in y is always constant. When our change in x is three, our change in y is always seven. So this is clearly a linear relationship. Now let's look at this one. Let's see. Looks like our x's are changing by one each time, so plus one. Now what are y's changing by? Here it changes by two. Then it changes by six. Alright, it's clearly not linear. Then it changes by 18. Clearly not a linear relationship. If this was linear, these would be the same amount, same delta, same change in y for every time, 'cause we have the same change in x. So let's test to see if it's exponential. If it's an exponential, for each of these constant changes in x's, when we increase x by one every time, our ratio of successive y's should be the same. Or another way to think about it is what are we multiplying y by? So to go from one to three, you multiply, you multiply by three. To go from three to nine, you multiply be three. To go from nine to 27, you multiply by three. So in a situation where every time you increase x by a fixed amount, in this case one, and the corresponding y's get multiplied by some fixed amount, then you're dealing with an exponential relationship. Exponential. Exponential relationship right over here.