Graph intersection of 3 curves

I have 3 functions:

x=-1:0.001:1;

y1=((3*x.^2)-1)/2;

y2=((5*x.^3)-(3*x))/2;

y3=((35*x.^4)-(30*x.^2)+3)/8;

plot(x,y1,'g-',x,y2,'r--',x,y3,'b-.');

I need to find all the interception points in the graphic.

The result should look like this:

Answer 1

[Star Strider]

This was a fun project!

y1 = @(x) ((3*x.^2)-1)/2;
y2 = @(x) ((5*x.^3)-(3*x))/2;
y3 = @(x) ((35*x.^4)-(30*x.^2)+3)/8;

e0 = linspace(-1,1,10);
for k1 = 1:length(e0)
    y1y2(k1) = fzero(@(x) (y1(x)-y2(x)), e0(k1));
    y1y3(k1) = fzero(@(x) (y1(x)-y3(x)), e0(k1));
    y2y3(k1) = fzero(@(x) (y2(x)-y3(x)), e0(k1));
end
y1y2 = unique(round(y1y2*1E+3)/1E+3);
y1y3 = unique(round(y1y3*1E+3)/1E+3);
y2y3 = unique(round(y2y3*1E+3)/1E+3);

x = linspace(-1,1);
figure(1)
plot(x,y1(x),'g-',x,y2(x),'r--',x,y3(x),'b-.');
hold on
plot(y1y2, y1(y1y2), 'kp', 'MarkerSize',8, 'MarkerFaceColor','y')
plot(y1y3, y3(y1y3), 'kp', 'MarkerSize',8, 'MarkerFaceColor','y')
plot(y2y3, y2(y2y3), 'kp', 'MarkerSize',8, 'MarkerFaceColor','y')
hold off

This might be written to be more efficient, but it works!

[J Neto]

Thank you so much, perfect solution!

[Star Strider]

My pleasure!

See Roger’s answer for (as usual) a more insightful solution. It was late and I honestly didn’t see that possibility.

Answer 2

[Roger Stafford]

Use 'roots' on the cubic polynomial difference between y1 and y2 to get the three intersections of the y1 and y2 curves. Similarly the difference between y1 and y3 and the difference between y2 and y3 will give quartic polynomials that produce four roots using 'roots'.