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So I have a quadratic Bézier curve which starts in $(1,1)$ and ends in $(0,0)$. It starts with a slope of $1/2$ and ends with a slope of $-1$. I want to determine its control points.

So first of all, since it's quadratic, it will only have one control point. But I am not sure how to find the control point.

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I have solved this problem now, the quadratic Bézier curve will be on some form $f(x) = a + bx + cx^2$, thus the derivative of the curve at the points $(1,1)$ and $(0,0)$ will be its tangents.

These tangents will meet at the control point of the quadratic Bézier curve which is easily determined since from the slopes and points we can determine their equations.

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