Math Family of curves Common point

# Common Point

When all functions of a family of curves go through a common point, a bundle of functions occurs.

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### Remember

Not every family of curves has a common point.
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### Hint

If there is a common intersection of all of the families function graphs, then there must be a location $x$ where the additional parameter disappears.
The intersection is therefore at this x-value.

### Example

$f_a(x)=x^2+ax$ (with $a\in\mathbb{R}$)

• $\color{blue}{f_3(x)=x^2+3x}$
• $\color{green}{f_1(x)=x^2+x}$
• $\color{red}{f_{-1.5}(x)=x^2-1.5x}$
For $f(0)=0^2+a\cdot0=0$, the parameter $a$ is eliminated at $x=0$. Therefore, the family of curves has a common point at $(0|0)$.