# How do I find the formula for a parabola from two points and $y$-coordinate for the minimum?

I am trying to find the function for a parabola. The things that are known are the points $$(x_1,y_1)$$ $$(x_3,y_3)$$ and in between somewhere the parabola has its vertex, which is its minimum. The $$x$$-coordinate of this point is not known only the $$y_2$$ is known. Now how am I supposed to find the function for the parabola? (I know that it is $$ax^2+bx+c$$ but how do I calculate the $$a$$,$$b$$ and $$c$$ from only the 2 and a half points?)

You have two equations:

$$y_1 = a{x_1}^2 + bx_1 + c$$ $$y_3 = a{x_3}^2 + bx_3 + c$$

and also know that the $$x$$-coordinate of the vertex is $$x_2 = -\frac{b}{2a}$$, so

$$y_2 = a{x_2}^2 + bx_2 + c$$

Substitute $$-\frac{b}{2a}$$ for $$x_2$$ in the above. You now have three equations whose unknowns are $$a$$, $$b$$ and $$c$$.

• Thanks for the answers. But could you explain how you know that what x_2 equals? Commented Nov 30, 2018 at 15:25
• Commented Nov 30, 2018 at 17:34

Another approach: Let $$y=a(x-x_2)^2+y_2\iff y-y_2=a(x-x_2)^2$$. Plugging in the coordinates and dividing both equations gives $$\frac{y_3-y_2}{y_1-y_2}=\frac{(x_3-x_2)^2}{(x_1-x_2)^2}.$$ Now convince yourself that the left hand side is positive, hence $$x_2$$ is easily calculated from $$\sqrt{\frac{y_3-y_2}{y_1-y_2}}=\pm\frac{(x_3-x_2)}{(x_1-x_2)}.$$