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## Algebra 1

### 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.
• this makes me want to cry and blow this up😍😍 • 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) • sup my homies want some stew • Hi , I was wondering if there is any relation between linear growth / exponential growth and the arithmetic/ geometric sequences ?
since exponential is the multiplication of the same number isn't that the same as the geometric sequence ?
Thanks ! • Good thinking! The answer is yes. An arithmetic sequence can be thought of as a linear function defined on the positive integers, and a geometric sequence can be thought of as an exponential function defined on the positive integers. In either situation, the function can be thought of as f(n) = the nth term of the sequence.
• Can someone explain to me why when you exponent a negative number, you don't a negative number every other time. • When we have `x^2`, that means we are calculating `x * x`. When we have `x^3`, that means we are calculating `x * x * x`.

Multiplying a negative number times another negative number results in a positive number.

If we have `-3^2`, that means we are calculating `-3 * -3`, which results in a positive number - 9.

If we have `-3^3`, that means we are calculating `-3 * -3 * -3`, which results in a negative number - -27.

When we have an even exponent, the result is a positive number. When we have an odd exponent, the result is a negative number.