DISCOVER: Graphs of Large Powers Graph the functions y=x^2, y=x^3, y=x^4, and y=x^5, for -1 ≤x ≤

DISCOVER: Graphs of Large Powers Graph the functions y=x^2,...

Key Concepts

Polynomial Functions

Polynomial functions are expressions involving sums of powers of x with constant coefficients. They are continuous, smooth, and their graphs are determined by the highest power (degree) of x, which influences the end behavior and the curvature of the graph. Understanding these functions is crucial when comparing the shapes of different power functions on a specific interval.

Even and Odd Functions

Even functions, which satisfy f(x) = f(-x), produce graphs that are symmetric with respect to the y-axis. This is typical for functions with even exponents, where negative inputs yield the same output as their positive counterparts. In contrast, odd functions satisfy f(-x) = -f(x), which means their graphs are symmetric with respect to the origin, a behavior observed in functions with odd exponents.

Behavior of High-Degree Power Functions on the Interval [-1, 1]

When considering power functions such as x^n on the interval [-1, 1], as the exponent n increases, the graph becomes increasingly 'flat' near x = 0 and steeper near the endpoints x = 1 and x = -1. For even exponents, the graph remains above or on the x-axis with symmetry about the y-axis, while for odd exponents, the graph passes through the origin and reflects origin symmetry. Very high powers, like 100 or 101, exaggerate these effects, making the transition near the endpoints much more abrupt.

Graphical Analysis and Use of Tables

Graphical analysis involves plotting functions and identifying key features such as intercepts and symmetry. Creating a table of values is a practical method to empirically observe function behavior, especially for confirming the rapid changes near the boundaries of the interval. This method helps to verify the expected flattening effect near x = 0 and the steep increases or decreases as x approaches 1 or -1 for large exponents.

Transcript

00:02 We just started to experiment around a bit with certain powers of graph looks like.

00:08 So i'm just going to be using a graphical calculator online, desmos.

00:13 It's very useful for this kind of thing.

00:15 It's free.

00:16 And i use it quite often.

00:18 It's very good.

00:20 I definitely recommend it.

00:22 So we're going to start off with x squared.

00:25 Everyone knows what that looks like.

00:26 Or you should do, at least.

00:28 It's very, very important.

00:30 X cubed another very very common graph x -parall 4 and we're also going to do xx4 and we'll leave it there for now and i'll just talk a little bit about this so you've got our x squared here very very normal and then we got our x the power of 3 here in the blue very normal but comparing that to the other two graphs so we got our purple here and for our x the 5 it looks very normal.

01:02 Very, very similar to the x of 3, except it's a bit more constrained, a bit more constricted.

01:08 And x of 4 is also very similar to x squared.

01:15 But it's asking us to look very specifically at the graphs between 1 and minus 1.

01:23 Okay, so that's roughly what we would see.

01:26 So we've got our graph here of x squared.

01:28 And it's sort of as you go further and further outwards, at least focusing, we'll focus on this bit for the moment.

01:35 As you go further and further outwards, you get, it goes further and further down and more towards a sort of straight line at the bottom, but then zooming towards the top after one.

01:49 So it comes closer and closer to this line.

01:52 And it asks the question, what do you think x the power of 100 would look like, or x the power of 101? so, you know, you've had your own think about it, but what i would think is that an x -fal of a 100 graph would look virtually almost like a straight line.

02:10 It'll sort of look like a really, really straight curve, sort of right down at the bottom, right near that x -axis.

02:18 And we'll just see if that's roughly what it looks like, yeah.

02:22 So it looks right like that up until it gets quite close to one.

02:28 And you can imagine that you can imagine the reason for this is that if you were to start plugging numbers in you'd get and we'll do that in a second we'll stop plugging numbers in if you start plugging numbers in you'll get as you approach one you're going to get if you to a power of 100 for example as you go further and further into the decimals it's going to be very very very small numbers very quickly because you can imagine the numbers decreasing every time you times it together.

02:59 So if you have 0 .8 and you times that together a hundred times, that's going to be a very small number.

03:04 Or if you have, even like 0 .98, it's still going to be less than one.

03:08 It's always going to be less than one...

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