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Algebra I (2018 edition)

Course: Algebra I (2018 edition)>Unit 13

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.

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• 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?
• 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`.
• 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?
• 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.
• At , how did Sal know to increase the function by 1.05?
• 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.
• Why is it worth comparing these two types of functions? Are linear and exponential models more important than other functions?
• 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.
• is the equation y=40(1.05)^i here i is the week, correct?
• Yes, that would be the equation for the 1st problem in the video.
• not really a question. I really like how the equation/ formula is explained in great detail.
• So linear functions increase by constant VALUES, while exponential functions increase by constant PERCENTAGES?
• 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.
• Can a Linear Function be expressed into an Exponential Function? In other words, Can you convert it into an Exponential Function? Thanks a lot.
• 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.
• how is that different from the geometric sequence? how would it be worded if it was a geometric sequence?
• 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!