Creating a Parabola that has a Curtain Gradient at a specific x coord

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wlayton

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In the image below we have a point at (3,8) that has gradient of -2 (imagine this point being the end of a linear function that has been limited as its in a piece wise function) and a parabola at with a turning point at (10,2).
1611999693005.png​
My goal is to make it so that the turning point of the parabola stays where it is but the dilation of it reaches the point (3,8) and also has the same gradient at that point as well. so that it is a smooth transition between the point (3,8) and the parabola. Is this do-able? and could someone help me and give me the process on how to do it.
Thanks :)
 
A point doesn't have a gradient; I suppose what you mean is that for x<3, you are imagining a line with that slope, which you didn't plot.

If you flatten the parabola, its slope at that point will be determined by the coefficient you use; you don't have separate control of it. That is, you have only one degree of freedom, which is "used up" in making the parabola pass through the given point.

To be specific, your parabola is y = a(x-10)^2+2. You can set x=3 and y=8 and solve for a; then you have nothing else to change in order to get the desired slope. You would have to add some additional freedom to the problem, such as not using a parabola.
 
Ok, Thanks. Ill see what i can do then. How about if i move the turning point of the parabola, is it possible then?
 
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