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

Graphs of Large Powers Graph the functions y=x^2 y=x^3,...

Key Concepts

Power Functions

Power functions of the form y = x^n exhibit behavior that strongly depends on the exponent. When |x| is less than 1, as the exponent n increases, x^n becomes very small, approaching zero. Conversely, when |x| equals 1, the value remains 1 (or -1 in magnitude), so the function 'pulls away' sharply near the endpoints. This behavior is central to understanding the graph of high power functions, particularly on the interval [-1, 1].

Even vs Odd Exponents

Even and odd power functions have distinct symmetry properties. Even exponents, like in y = x^100, produce graphs that are symmetric about the y-axis because (?x)^n = x^n when n is even. This results in both sides of the graph behaving the same way, with the function being near zero for most of the interval and rising steeply near the endpoints. In contrast, odd exponents, such as y = x^101, yield graphs that are symmetric about the origin since (?x)^n = ?x^n when n is odd, causing the function to preserve the sign of x. This leads to a graph that is very flat near the origin but rises (or falls) steeply as x approaches 1 (or -1).

Graph Behavior on the Interval [-1, 1]

When graphed on the interval [-1, 1], high even power functions appear nearly flat, hovering very close to zero over most of the interval except near x = 1 and x = -1, where they quickly jump to 1. High odd power functions, meanwhile, remain close to zero for values of x well away from ?1 and 1 but cross through the origin and then, near the endpoints, rapidly increase or decrease. This highlights the dramatic effect that high exponents have on graph shape in a bounded interval.

Transcript

00:03 When we talk about the graphs, the power functions, they tend to have similar characteristics.

00:10 So in other words, the graph of y equals x squared is going to be a parabola with the vertex at the origin.

00:21 Y equals x to the fourth power is going to be the same kind of function.

00:28 What's going to happen is it's just going to flatten out a little bit more as we get close to zero.

00:33 But they're both going to have the coordinate, negative 1 -1 and positive -1 -1, in common.

00:40 So the higher the power, the more flat we get in the middle.

00:43 So the graph of y equals x to the 100th power is going to have still the same characteristics, but it's just going to get really flat as we get towards the middle here between negative 1 and positive 1.

01:01 Similarly, all of the odd functions look the similar, to y equals x cubed.

01:09 We know our cubic function does something like this.

01:13 Goes through the coordinate negative 1, negative 1, and positive 1, positive 1.

01:18 Y equals x to the fifth power is going to come up and go through those same two coordinates, but just get a little bit flatter in the middle.

01:29 So the graph of y equals x to the 100 first power is going to be just even more flat in the middle.

01:40 And we can display that a little bit easier we can demonstrate this with desmos.

01:47 If we have y equals x squared, we can see here what we're looking at.

01:57 In fact, if we wanted to see what was happening between negative one and positive one, i'll set the graph, so we're going from negative two, positive two.

02:11 So we go a little bit beyond that.

02:14 But here we have the coordinate, negative 1, 1, and positive 1, positive 1...

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