Curve sketching of family of curves

Even with family of curves you can apply curve sketching.

Example

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

• Zeros
• Extrema
• Inflection points

1. Find the derivative

   $f_a(x)=x^2+ax$
   $f_a'(x)=2x+a$
   $f_a''(x)=2$

2. Zeros

   calculating Zeros: Set function equal to zero
   $f_a(x)=0$
   $x^2+ax=0$
   $x(x+a)=0$
   $x_{N_1}=0$ and $x_{N_2}=-a$

3. Extrema

   calculate extrema: Set the first derivative equal to zero
   $f_a'(x)=0$
   $2x+a=0\quad|-a$
   $2x=-a\quad|:2$
   $x_E=-\frac{a}2$
   
   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:
   $f_a(-\frac{a}2)$ $=(-\frac{a}2)^2+a\cdot(-\frac{a}2)$ $=\frac{a^2}4-\frac{a^2}2$ $=\frac{a^2}4-\frac{2a^2}4$ $=-\frac{a^2}4$
   
   $T(-\frac{a}2|-\frac{a^2}4)$

4. Inflection points

   calculate inflection point: Set second derivative equal to zero
   $f_a''(x)=0$
   $2=0$ => function has no inflection points