Plot the expressions -x^4 and (-x)^4 on the same graph in the window -1<x<1,-1<y<1. Is -x^4=(-x)

Plot the expressions -x^4 and (-x)^4 on the same graph in...

Key Concepts

Order of Operations with Negatives

This concept emphasizes how the placement of the negative sign determines the result of an expression. In mathematics, the negative sign applied outside an exponent, as in -x^4, means that the exponent is computed first and then the sign is applied, resulting in -(x^4). Conversely, when the negative sign is inside the parentheses, as in (-x)^4, the negative is part of the base being raised to the power, which may lead to a different outcome, particularly when the exponent is even.

Graphing Functions and Comparative Analysis

Graphing is a useful tool for visually analyzing and comparing functions. When plotting functions such as -x^4 and (-x)^4 over a given interval, one can directly observe their behavior and differences. This approach helps in understanding how modifications in the algebraic form of an expression can lead to significant differences in the shape, symmetry, and orientation of the graph.

Even Exponents and Function Symmetry

Even exponents have the property that raising both a positive and its corresponding negative value produces the same result. This means that in the expression (-x)^4, the negative sign disappears when raised to the fourth power, yielding the same output as (x)^4. This property of even functions results in symmetry about the y-axis and is crucial when comparing expressions like -x^4 and (-x)^4.

Transcript

00:01 All right, so we're asked to find the graph to the right of x being greater than negative 1.

00:08 So that's all i'm doing right now is showing the graph to the right of x equals negative 1.

00:16 And you probably remember in a pre -couculous, there's a lot of tricks to doing this.

00:24 For instance, if you look at 4x over x plus 1 squared, that you can set the numerary number.

00:32 Equal to 0.

00:34 If you set the numerator equal to 0, that will find, well, the y value equals 0.

00:41 So that only happens when x equals 0, because you can divide the 4 over.

00:45 So x equals 0.

00:47 So the ordered pair 0 is on the curve.

00:51 If you set the denominator equal to 0, change colors, that'll find you the vertical asymptote, which is exactly happening there.

01:00 Now what you don't know is what kind of curve you have.

01:05 Let me make sure.

01:08 So what they want you to do is to take the derivative of that, which would be your quotient rule.

01:14 So they had this as y equals.

01:16 So d, y, d, x, is equal to the derivative of the top, leave the bottom alone, minus the derivative of the bottom, which is your chain rule.

01:29 X plus 1 is now to the first power.

01:30 Leave the top alone.

01:32 All over the denominator.

01:33 It is squared.

01:36 So x plus one would be to the fourth power because squared squared is to the fourth.

01:40 Now what i would do is i would try to simplify a little bit.

01:44 For instance, every one of these terms has an x plus one.

01:48 So i can now take out an x plus one and cancel it out.

01:52 So we're looking at four...

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