introduction

As a bit of a geek math nerd, I spent days leading up to Valentine’s Day trying to find a math function appropriate to the day. In the end, I did not find a suitable individual function, but (perhaps aptly enough) a pair: function and inverse function. It occurred to me that these would be a great opportunity to illustrate a series of simple math transformations involving reflections on the x-axis, y-axis, and both the x-axis and y-axis. While the inverse function itself represents the reflection in the line y = x.

The function (and its inverse)

So here is the pair of graphs I initially committed to: $$ y = -x (x-4) $$ and $$ x = -y (y-4). $$ As can be seen, there is an appearance of a heart shape formed by the function and its inversion. Maybe not quite what Valentine card designers would be looking for!

Heart shape formed by the function and its inversion

I was fascinated to find that this particular function and its inverse have 4 intersections. Two of them can be solved by simple inspection. For the other pair, since it is a quadratic and a quadratic root function, we have to solve a polynomial of order 4: $$ y = -x (x-4) = y (y-4) (- y (y- 4) -4), $$ or $$ – y ^ 4 + 8y ^ 3-20y ^ 2 + 15y = 0 $$. Since such a problem is beyond my mathematical reach, I submitted the equations (as a simultaneous pair) to Wolfram Alpha to come up with some interesting solutions:

Function and its inversion have 4 points of intersection

From the diagram it should be clear that we expect the solutions to be symmetrical about the line y = x, so it is interesting that WA represents the y values ​​in a somewhat complicated form. The y-value of the first solution should match the x-value of the second and vv (and this is easily verified by the reader).

reflections

Typical school-level operations for functions are reflections on the x-axis (y -> -y) and reflections on the y-axis (x -> -x). For the reflection in the x-axis, the corresponding equations (function and inverse) are: $$ y = x (x-4) $$ and $$ x = y (y-4). $$ For the reflection in the y-axis: $$ y = -x (x + 4) $$ and $$ x = -y (y + 4). $$ To complete the amount of reflection we need to reflect in the x-axis and then in the y-axis or vv (both should give the same result): y = x (x + 4) or x = y (y + 4). Now let’s plot all 4 functions and their inversions with the Desmos online graphics tool:

Draw all 4 functions and their inversions

As can be seen, we get a “design” that has several different lines of symmetry. In addition to the above, we seem to have some “inherited” symmetry about the line y = -x, although this was not a specific part of the design process described above. The reader can see that the graphic contains both large and small hearts. For example, the parabola with an inflection point (-2; 4) in the second quadrant shows a corresponding inflection point of the inverse function (4; -2) in the fourth quadrant. Each function / inverse pair has 4 points of intersection, 2 of which lie on the line y = x, as can be seen visually. The other pair is obtained by solving a polynomial order 4, as we have explained and illustrated earlier (the solution theorem contains the pair on the line y = x which leads to 4 intersections, as we would expect from solving a ‘quartic’ equation could). .

y = -x (x-1) and derivatives

This function set indicates either one or two intersections for each function / inverse pair. All of them (as for the set -x (x-4)) have one solution at (0; 0) and the second solution (s) (for the parabolas with inflection points in quadrants 2 and 4 are at (2; 2 )) and (-2; -2). The parabolas with inflection points in quadrants 1 and 3 only intersect their inverse function partners at (0; 0). The graphic shows a similar symmetry as the first one, but (at least to the author’s eye) has a “cleaner” appearance as there are fewer intersections. A very “unromantic” perception, perhaps because this graphic does not contain hearts!

Function set shows either one or two intersection points

y = -x (x-3) and derivatives

In this set of functions, the parabolas with inflection points in quadrants 1 and 3 intersect their inverse functions only along the line y = x at points (2; 2) and (-2; -2), respectively. As usual, all parabolas / inverses have an intersection at the origin. The parabolas with inflection points in quadrants 2 and 4 have four points of intersection, the inversions of which lead to “hidden” heart shapes in the following graphic. For example, consider the parabola with an inflection point at (1.5; -2.25) and its inverse with an inflection point at (-2.25; 1.5). The function meets the inversion at (0; 0) and (4; 4) and forms the heart shape. As the reader traces these curves, it becomes clear that there are two more intersections.

Parabolas with inflection points in quadrants 1 and 3 only intersect their inverse functions

Heart condition

We can ask the question under what special circumstances we get heart-shaped graphics that are different from others that do not have such properties. Mathematically we ask when graphs of the form y = -x (xk) 4 intersect with their corresponding inverse functions. To answer this question, let’s send Wolfram Alpha a verbatim equation – by requesting a solution for the simultaneous pair y = -x (xk) and x = -y (yk):

simultaneous pair y = -x (xk) and x = -y (yk)

The above solution contains a square in k under the square root sign. Therefore we determine where there will be real solutions (apart from x = 0 and x = k-1) by solving the inequality: $$ k ^ 2-2k-3 ge 0 $$. Fortunately, the square easily factories itself as (k-3) (k + 1) and so we quickly get the solution: $$ k ge 3, $$ or $$ k le {-1}. $$ It should now be obvious why we have chosen values ​​of k = 3 and k = 1 for two of our examples above. These are limits of the square above.

We have to note here that technically the limit value is k = -1, which corresponds to the equation y = -x (x + 1) and its inverse x = -y (y + 1). However, this pair is in the ‘set’ of functions for which the ‘base pair’ is y = -x (x-1) and x = -y (y-1). In this case k = 1, so we are in the area where the only intersection points are on the line y = x. In fact, only corresponds to (0; 0), since (according to the general solution above) the other solution to (k-1; k-1) also corresponds to (0; 0) when k = 1.

With k = 3, two of the “sets” actually have a k-value of -3 and therefore fall within the range in which we expect 4 intersections and heart shapes. As already described, these “hidden” hearts can be found in the graph which shows k = 3 and derivatives.

y = -x (x-0.5) and derivatives

In this case the k-value is either 1/2 or -1/2 and both are in the range where we don’t expect the graphs to catch their inverses, except along the line y = x. In fact, this is true for all fractional values ​​of k – k = 1/2 is not particularly unique.

Graphs for intercepting their inverse except along the line y = x

Special case: k = 2 and derivatives

The graphs of interest here are those of y = -x (x + 2) and y = x (x-2), the turning points of which are in the second and fourth quadrants, respectively. The corresponding inverse functions are x = -y (y + 2) and x = y (y-2). The k-value is -2, so we expect 4 intersections between function and inverse. When we ask Wolfram Alpha to solve these points in the case of y = x (x-2) and inversely, we find (remarkably) that the solution pair is nothing but the “golden ratio” and its negative reciprocal. We can quickly check that this pair satisfies both equations by using the unique ‘algebra’ of the golden ratio. ## phi. ## Given ## x = phi: ## $$ y = phi ( phi-2) = phi ( phi-1-1) = phi ( phi ^ {- 1} -1) = 1- phi = – phi ^ {- 1} $$. ## y = – phi ^ {- 1}: ## $$ x = – phi ^ {- 1} (- phi ^ {- 1} -1-1) = phi ^ {- 1} ( phi ^ {- 1} + 1 + 1) = phi ^ {- 1} ( phi + 1) = 1 + phi ^ {- 1} = phi $$. Another pair of solutions is the same except that x and y values ​​are swapped since the solutions are symmetrical about the line y = x.

y = -x (x + 2) and y = x (x-2)

Summary and conclusion

In this article we examined different sets of functions based on the ‘generic’ function y = -x (xk). While the “search” for heart-shaped graphics seems a little frivolous, it has produced some interesting mathematical considerations regarding the determination of the intersection of parabolas and their inverse functions. We have also determined the special conditions under which the characteristic heart-shaped graphics are created. We hope the article has at least some good ideas for student exercises involving functional reflections and inversions.

Since the article started with a heart-shaped proto-design, we’ll end with one that incorporates various aspects of the theory discussed in the article. Namely the set of graphs based on y = -x (x-2). We leave it up to the reader to judge whether or not this final graphic will cut ice with Valentine card designers!

Graph based on y = -x (x-2)

References

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