Look at the graph of x^2
R.036035c45a2ac01824e8245c46f1a534 (546×314) (bing.com)
a nice “V” shape.
here’s the graph of x^3
R.c6f80415c6f0a653349cdccae3032821 (1002×821) (bing.com)
An “S” shape
Essentially, any even exponent will be a V, and any odd exponent will be an S.
How does the negative X’es graph when the exponent is NOT a neat odd nor even integer?
In general you can’t graph y = x ^ (m/n) for the negative values of x in a meaningful way other than a few special cases where it is well defined (integer exponents or reduced fractional exponents where n is odd).
If you allow complex operations then you can get some meaningful answers for the general expression.
ETA: You can play around with it here: Desmos | Graphing Calculator
Try x^(5/3) and then x^(5/2)
Just treat it as an algebraic variety, e.g. plot points where y^5-x^3=0 or y^4-x^3=0 or y^2-x^5=0.
E.g., y^2=x^5 has a higher-order cusp at the origin; if you graph it (e.g Equation Grapher) you will see what I mean—it is “pointy” (singular)
To look at y=x^a for different values of a, try this:
I see that @Jas09 beat me to it with a Desmos link, but mine should have the equation already typed in.
For any fractional exponent with an even denominator, the negative part of the graph would go imaginary. To properly graph imaginary numbers, you’d need four-dimensional graph paper, though one does occasionally encounter various workarounds, depending on what one wants to show with the graph.
For a simple example, y=x^\frac{1}{2} is the same as y=\sqrt{x}. And of course we all know that the square root of a negative number is imaginary.
A way of visualizing it:
Consider the graph for y=x^a as a goes from 2 to 3. When a=2, the tail points up, and when a=3, it points down.
But between those points, there must have been a continuous motion–the tail must have whipped around somehow. The points obviously didn’t go through zero. Instead, the tail whips around in the complex plane.
Not 100% sure this will work, but see this animated graph:
The tail flips up and over as a goes from 2 to 3. Same thing for higher powers.
It occurred to me, for graphing negative values of x, one need only be restricted to the Absolute Value of the solution.
However, when I attempt to use brackets …[ and]…as I was taught in math class many moons ago, I don’t get my desired result.
Anybody have a clue how to formulate this into those graph sites?
Wolfram Alpha does a pretty good job of making a visual.
Neither Desmos nor GeoGebra seem to properly handle complex numbers in their standard graphing calculators. They support absolute values (via |x^a|), but the complex part of it isn’t being handled. Not sure why. Math3D works properly, though.
For absolute value, you don’t use [brackets], you use |absolute value bars|. (On my keyboard, that’s Shift-Backslash.)
And thus the solution is revealed!
Thanks to Master Boink!
Though you can get around this by putting the absolute value inside the base, instead of outside the exponentiation: |x|^a.
