I have assumed that the equation is f(x)=kx * (1-x)
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 [0.1] but you can ignore everything below the X-axis).
![enter image source here]()
As you can see as k increases in value the vertex rises and the sides get "squeezed" in.
For the interval [0,1] you are probably more interested in the vertex.
If f(x)=k * x * (1-x)
color(white)("XXX")=kx-kx^2
color(white)("XXX")=(-k)(x^2-x)
color(white)("XXX")=(-k)(x^2-x+1/4)+k/4
color(white)("XXX")=(-k)(x-1/2)^2+k/4
which is the vertex form of a parabola with vertex at (1/2,color(red)(k/4))
So we can see that the x coordinate of the parabola stays constant but the y coordinate varies as 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.
