Math Family of curves Curve sketching of a family of curves

Curve sketching of family of curves

Even with family of curves you can apply curve sketching.



In curve sketching of family of curves, one often works with solutions that also include the additional parameter.


When taking derivatives of family of curves, the additional parameter must be treated as a constant.


Examine $f_a(x)=x^2+ax$ ($a\in\mathbb{R}$) for the following properties:

  • Zeros
  • Extrema
  • Inflection points
  1. Find the derivative

  2. Zeros

    calculating Zeros: Set function equal to zero
    $x_{N_1}=0$ and $x_{N_2}=-a$
  3. Extrema

    calculate extrema: Set the 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:
    $f_a(-\frac{a}2)$ $=(-\frac{a}2)^2+a\cdot(-\frac{a}2)$ $=\frac{a^2}4-\frac{a^2}2$ $=\frac{a^2}4-\frac{2a^2}4$ $=-\frac{a^2}4$

  4. Inflection points

    calculate inflection point: Set second derivative equal to zero
    $2=0$ => function has no inflection points