# Graphing functions with non-integer exponents **URL:** \ **Category:** Factual Questions\ **Created:** [January 11, 2024, 2:52pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951 "2024-01-11T14:52:44Z")\ **Posts on this page:** 12\ **Page:** 1
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [January 11, 2024, 2:52pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/1 "2024-01-11T14:52:44Z")
Look at the graph of x^2 [R.036035c45a2ac01824e8245c46f1a534 (546×314) (bing.com)](https://th.bing.com/th/id/R.036035c45a2ac01824e8245c46f1a534?rik=X5KbL8e4sP9IDQ&riu=http%3a%2f%2fwww.bbc.co.uk%2fschools%2fgcsebitesize%2fmaths%2fimages%2fgraph_24.gif&ehk=Zm0cEt%2bBmsUUuNnJrzcP7iNLF5pTJ3tTpZVme%2bvkiMU%3d&risl=&pid=ImgRaw&r=0) a nice “V” shape. here’s the graph of x^3 [R.c6f80415c6f0a653349cdccae3032821 (1002×821) (bing.com)](https://th.bing.com/th/id/R.c6f80415c6f0a653349cdccae3032821?rik=Fa0LPmTGu8Q9Rg&riu=http%3a%2f%2fmyarnell.diymaths.com%2fPictures%2fcubicgraph.jpg&ehk=RhOyzXOJ7xzp%2b4ZZJSK4Dsebtk4YYz%2b8vuqlOt1x8%2f0%3d&risl=&pid=ImgRaw&r=0) 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? ---
**Author:** ![Jas09](https://avatars.discourse-cdn.com/v4/letter/j/d07c76/32.png) [@Jas09](https://boards.straightdope.com/u/Jas09)\ **Post date:** [January 11, 2024, 3:05pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/2 "2024-01-11T15:05:57Z")
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](https://www.desmos.com/calculator) Try x^(5/3) and then x^(5/2) ---
**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\ **Post date:** [January 11, 2024, 3:57pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/3 "2024-01-11T15:57:35Z")
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](https://www.mathsisfun.com/data/grapher-equation.html)) you will see what I mean—it is “pointy” (singular) ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 11, 2024, 4:04pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/4 "2024-01-11T16:04:59Z")
To look at y=x^a for different values of a, try this: > **[Desmos | Graphing Calculator](https://www.desmos.com/calculator/qmyyl4tlka)** > > Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. I see that @Jas09 beat me to it with a Desmos link, but mine should have the equation already typed in. ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 11, 2024, 10:46pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/5 "2024-01-11T22:46:39Z")
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. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 12, 2024, 12:43am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/6 "2024-01-12T00:43:13Z")
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: > **[Math3d: Online 3d Graphing Calculator](https://www.math3d.org/nxcsSrvrA)** > > An interactive 3D graphing calculator in your browser. Draw, animate, and share surfaces, curves, points, lines, and vectors. The tail flips up and over as a goes from 2 to 3. Same thing for higher powers. ---
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [January 12, 2024, 1:14am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/7 "2024-01-12T01:14:38Z")
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? ---
**Author:** ![Chingon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chingon/32/17645_2.png) [@Chingon](https://boards.straightdope.com/u/Chingon)\ **Post date:** [January 12, 2024, 1:15am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/8 "2024-01-12T01:15:49Z")
Wolfram Alpha does a pretty good job of making a visual. ---
**Author:** ![Dr.Strangelove](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/dr.strangelove/32/6613_2.png) [@Dr.Strangelove](https://boards.straightdope.com/u/Dr.Strangelove)\ **Post date:** [January 12, 2024, 1:20am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/9 "2024-01-12T01:20:26Z")
> [@EnolaStraight](#): > > Anybody have a clue how to formulate this into those graph sites? 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. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [January 12, 2024, 1:21am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/10 "2024-01-12T01:21:41Z")
> [@EnolaStraight](#): > > when I attempt to use brackets …[and]…as I was taught in math class many moons ago For absolute value, you don’t use [brackets], you use |absolute value bars|. (On my keyboard, that’s Shift-Backslash.) ---
**Author:** ![EnolaStraight](https://avatars.discourse-cdn.com/v4/letter/e/41988e/32.png) [@EnolaStraight](https://boards.straightdope.com/u/EnolaStraight)\ **Post date:** [January 12, 2024, 1:44am UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/11 "2024-01-12T01:44:03Z")
And thus the solution is revealed! Thanks to Master Boink! ---
**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\ **Post date:** [January 12, 2024, 2:21pm UTC](https://boards.straightdope.com/t/graphing-functions-with-non-integer-exponents/995951/12 "2024-01-12T14:21:19Z")
> [@Dr.Strangelove](#): > > They support absolute values (via |x^a||xa||x^a|), but the complex part of it isn’t being handled. Though you can get around this by putting the absolute value inside the base, instead of outside the exponentiation: |x|^a.