In the Long Run 🏃💨


After completing the Desmos activity. The two functions I have chosen to elaborate on in this blog are y=(x-2)^4 and  y=2(x+3)^7. The reason why I picked these two functions are how the two graphs become one graph that appears to be aligning with the y-axis; when zoomed out. However when an individual zooms into the graph. The function y=(x-2)^4 is actually a parabola. As for the function y=2(x+3)^7 is a chair shaped function that rests vertically and extremely close to the y-axis.  The short run (being the behavior of the graph close to the origin) of the function y=(x-2)^4, the value is positively increasing. As for the function y=2(x+3)^7 the graph starts off on the negative side of the x-axis. The long run (the behavior of the graph close to the ends) of the two graphs are both positive.

The reason why the relationship differs depending on how zoomed in or out the graph of these two functions. It deals with small and large values. We are used to this causing functions to cross paths in a linear function however in the case of polynomials. The majority of the time graphs are shown merging into one function as the values increase.

I have attached screenshot of this being shown.

Screen Shot 2017-03-26 at 11.08.13 PMScreen Shot 2017-03-26 at 11.04.11 PM

To begin explaining a polynomial.  The phrase “poly” stands for many and the phrase “nomial” stands for terms. So it’s no surprise that the definition is basically many terms. They consist of constants (which I will explain later), variables and exponents. Examples are the following equations.

  • 7x^3+8x- 9
  • 2x^5  and 4x
  • 4(x^3-7x^4)
  • 2(6x^2)+7x-x+8x^3
  • -0.05x^3+ 3x+ 12

Using the equation from Desmos: f(x)=4x^3-2x^7+2x+15. I will now explain what is a constant term, leading term and a degree.

A constant term is numbers like  15, because it stands on it own. A leading term is -2x^7, because it contains the highest power.  While the degree is 3, 7, 1, and 0; because it is the sum of the exponents of the variables.

Two types of functions we’ve studied already that are polynomials are quadratic functions and power functions.

Can you figure out how the short-run and long run are used in real life scenarios? One example I have is the use of a prescribed drug on a patient over time.



One Comment Add yours

  1. Dr. Fisher says:

    That dolphin gif is trippy!

    This was a great post. I really like where you said that “The reason why the relationship differs depending on how zoomed in or out the graph of these two functions. It deals with small and large values.” The key thing to note is that when zoomed out, you’re looking at large x values like x = 1,000,000. If you have a large exponent like x^5 then this number will be (1 million)(1 million)(1 million)(1 million)(1 million). If a different term like 6x has a small exponent then this would just be 6 million. Thus, when you zoom out, the terms with large exponents overpower the other ones and become all that is really visible in the graph.


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