On Graphing Polynomials, And The Behaviour Of Such Graphs At Their Ends

Question

I'm learning about graphing polynomial functions currently. The textbook I'm reading from says that no matter the polynomial function, the final behaviour of the ends of a polynomial graph will always be the same as the end behaviour of the graph of the polynomial in it with the highest degree. For instance, according to the textbook, if I plotted $y=x^{10}+x^{3}+10x^{2}$, the end behaviour of it would resemble $y=x^{10}$.

Can someone please explain to me why this is the case? I suspect it is because as $x$ becomes bigger, the value of $x^{10}$ becomes so big that it makes the effect of the other values insignificant.

But still, I'm not sure about this. With a huge number, wouldn't the difference in distance between the 2 graphs also be bigger than ever before? So the 2 graphs will only look similar when you zoom out from the 2 of them, in which case, if you zoom at far enough, all graphs will look the same. So if a huge number is inputted, and the distance between the 2 graphs is greater than ever before, how can we still say that with a huge number, the 2 graphs will begin to look and behave the same?

Lastly, if possible, try to explain it as simply as possible, without Calculus, because I'm still a beginner.

Answer 1

Keep in mind that when your text speaks of "the final behaviour of the ends of a polynomial graph" it speaks of only three characteristics of the ends of the graph

1. whether the graph is positive or negative
2. whether the graph is increasing or decreasing
3. whether the graph is concave up or concave down

All other properties of the ends of the graphs are omitted from consideration.

It is solely in this regard that the behaviour of a polynomial graph is the "same" as the behaviour of its highest degree term.

Answer 2

Just as in your question, let us define $$ f(x) = x^{10} + x^{3} + 10x^{2}, \hspace{20pt} g(x) = x^{10}. $$ We will explore the behaviour of these two graphs. I will leave it to you to graph them (Desmos is a great site for graphing functions) and to play around with different functions to see how their limiting behaviour changes.

Firstly, you write the following:

With a huge number, wouldn't the difference in distance between the 2 graphs also be bigger than ever before?

This is indeed true! For $f$ and $g$ above, the difference between them is $$ f(x) - g(x) = x^{3} + 10x^{2}, $$ and this does increase as $x$ increases. Thus it is true that $f$ and $g$ do not get closer and closer as $x$ gets larger --- but this is not what limiting behaviour describes. Instead of the additive difference between $f$ and $g$, limiting behaviour is more concerned with the multiplicative difference of $f$ and $g$, namely the quotient $f(x)/g(x)$. (For Mathemticians, the formal way to describe limiting behaviour is to use Big O notation, but this may be slightly out of your reach at the moment.)

Let us plot some values to see this explicitly (values are to $4$ decimal places).

$$ \begin{array}{cccc} x & f(x) & g(x) & f(x)/g(x) \\ \hline 1 & 12 & 1 & 12 \\ 2 & 1072 & 1024 & 1.0469 \\ 3 & 59166& 59049& 1.0020 \\ 4 &1048800&1048576&1.0002 \\ \end{array} $$

Just by comparing $f(x)$ and $g(x)$ for the larger values, they look "more or less the same", despite the actual difference of $f(x)$ and $g(x)$ increasing.

Let us try a more extreme example. Define $$ f(x) = x^{10} + 999x^{9}, \hspace{20pt} g(x) = x^{10}. $$ Let us plot some values to see this explicitly.

$$ \begin{array}{cccc} x & f(x) & g(x) & f(x)/g(x) \\ \hline 1 & 1000 & 1 & 1000 \\ 2 & 512512 & 1024 & 500.5 \\ 3 & 19722366 & 59049 & 334 \\ 4 & 262930432 & 1048576 & 250.75 \\ 5 & 1960937500 & 9765625 & 200.8 \\ 10& 1009000000000 & 10000000000 & 100.9 \\ 100&1099000000000000000000&100000000000000000000&10.99\\ \end{array} $$

Again, the values seem to get more and more similar as $x$ increases (I suggest doing this yourself with some very large numbers to see how similar they get), and the multiplicative difference between the two is decreasing.

In fact, in both cases the multiplicative difference $f(x)/g(x)$ is getting closer and closer to $1$, which must mean that $f(x)$ and $g(x)$ are getting 'more similar' in some sense. You can see this algebraically: for $x$ sufficiently large, we have $$ \frac{f(x)}{g(x)} \approx 1 \implies f(x) \approx g(x). $$ In other words, the difference between the functions becomes negligible, since the numbers themselves are getting bigger far quicker than the difference between them is.

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If you want to evaluate these large polynomials yourself, I suggest using Python (this is my personal preference since it can deal with very large numbers easily), and the corresponding code that I used to to define the functions is:

def f(x):
  return x**10 + 999*x**9

def g(x):
  return x**10

You then evaluate each polynomial by writing f(1), for example. Note that the double-asterisk ** is the Python code for exponents, so use ** in place of ^.