# Question #c1efc

Mar 28, 2017

$k$ is the number which after division by $4$ is the $y$ coordinate of the vertex.

#### Explanation:

I have assumed that the equation is $f \left(x\right) = k x \cdot \left(1 - x\right)$
and you want to know the effect of varying the value of $k$.

Here is a graph with sample values of $k$ (I did not limit it to the interval $\left[0.1\right]$ but you can ignore everything below the X-axis).

As you can see as $k$ increases in value the vertex rises and the sides get "squeezed" in.

For the interval $\left[0 , 1\right]$ you are probably more interested in the vertex.
If $f \left(x\right) = k \cdot x \cdot \left(1 - x\right)$
$\textcolor{w h i t e}{\text{XXX}} = k x - k {x}^{2}$

$\textcolor{w h i t e}{\text{XXX}} = \left(- k\right) \left({x}^{2} - x\right)$

$\textcolor{w h i t e}{\text{XXX}} = \left(- k\right) \left({x}^{2} - x + \frac{1}{4}\right) + \frac{k}{4}$

$\textcolor{w h i t e}{\text{XXX}} = \left(- k\right) {\left(x - \frac{1}{2}\right)}^{2} + \frac{k}{4}$

which is the vertex form of a parabola with vertex at $\left(\frac{1}{2} , \textcolor{red}{\frac{k}{4}}\right)$

So we can see that the $x$ coordinate of the parabola stays constant but the $y$ coordinate varies as $\frac{1}{4}$ of the value of $k$ with changing values of $k$.

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Hope this is in some way related to what you were looking for.