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How do I graph the negative portion of a square root plot?

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I would appreciate any help possible in understanding how to get this code to graph the portion of each y function shown that comes from -1 to 0. As can be seen the portion shown for each y function graphs the part from 0 to 1 but not -1 to 0.
Voss on 9 Apr 2024
Moved: Voss on 9 Apr 2024
When x<0, y is complex (note the warnings when plotting); thus the real part of y is plotted.
%Question 11
x = -1:0.01:1;
y = sqrt(1 - 3.*x.^(2/3) + 3.*x.^(4/3) - x.^2);
Warning: Imaginary parts of complex X and/or Y arguments ignored.
% ylim([-1,1])
hold on
y = -y;
Warning: Imaginary parts of complex X and/or Y arguments ignored.

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Answers (1)

John D'Errico
John D'Errico on 9 Apr 2024
Edited: John D'Errico on 17 Apr 2024
Since in reality, you have what is called an implicit function, we want to use fimplicit. I might do it like this:
fun = @(x,y) (1 - 3.*nthroot(x,3).^2 + 3.*nthroot(x,3).^4 - x.^2) - y.^2;
That is, you will get the negative branch of the curve by squaring the square root. fimplicit does the heavy lifting for you here.
grid on
xlabel 'X'
ylabel 'Y'
Note my use of nthroot protects me from having problems when x is negative. This avoids creating complex numbers. For example, the real cube root of -2 is
ans = -0.7937
Therefore, the real 2/3 power of -2 can be found by squaring that result.
ans = 0.6300
The problem of course is if you simply use direct fractional powers, then MATLAB will return ONE of the solutions, but it will choose by default a complex solution for that fractional power. In fact, there are multiple fractional powers we must worry about as solutions. This is only one of them:
ans = -0.3150 + 0.5456i
I do something similar to deal with the 4/3 power of x. As well, squaring both sides allows fimplicit to plot all 4 branches, for both positive and negative x, as well as positive and negative y.
A bit tricky I'll admit, but, it works, and works correctly, because it deals properly with those nasty fractional powers.
fimplicit is a tremendously useful tool when used carefully.


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