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.

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### Hint

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

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

### 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