Math Family of curves Locus curve

Locus curve

The extrema or inflection points of all function graphs of a family of curves lie on a new graph, the locus curve.



In general, locus is a set of points satisfying some condition.


In this case, the locus curve is a new function, on whose graph there is a specific point (e.g. extrema or inflection point) of each function of the family of curves.

The picture: All minima of the family of curves $f_a$ lie on the graph of a new function $g$.


  1. Calculate searched points (e.g. extrema or inflection points)
  2. Set up equations for x and y values
  3. Rearrange equation for x-value and insert


Determine the function of all minima of $f_a(x)=x^2+ax$ ($a\in\mathbb{R}$)

  1. Find derivative

  2. Calculate minimum

    calculate extrema: Set first derivative equal to zero

    use suspicious points for extrema in the second derivative test:
    $f_a''(-\frac{a}2)=2>0$ => minimum

    calculate the y-coordinate and specify the minimum:
    $f_a(-\frac{a}2)$ $=(-\frac{a}2)^2+a\cdot(-\frac{a}2)$ $=\frac{a^2}4-\frac{a^2}2$ $=\frac{a^2}4-\frac{2a^2}4$ $=-\frac{a^2}4$

  3. Set up equations for x and y values

    from the coordinates of the minimum you can now set up two equations
  4. Rearrange equation and insert

    the equation for x is now changed to the parameter $a$ and inserted into the second one.

    $y=-\frac{(-2x)^2}4$ $=-\frac{4x^2}4$ $=-x^2$

    function of all minima: $y=-x^2$