Find quadratic equation given two points and y-intersection

Question

Let the domain be $x \in [0,h]$. We have three points, $(0,1)$, $(h/2,0)$ and $(h,0)$. How do I find the quadratic equation?

My attempt: I know that the roots are located at $x=h/2$ and $x=h$. Thus the equation must be of the form $(x-h/2)(x-h) = 0$.

Expanding out we have $x^2 - (h/2)x - hx + h^2 /2 =x^2 - (3h/2)x + h^2 /2 = 0$.

I'm not sure what to do with the y-intersection.

Answer 1

You are close. The quadratic is of the shape $y=A(x-h/2)(x-h)$, where $A$ is a constant.

Use the point $(0,1)$ to identify $A$.

Answer 2

The equation must be of the form $a(x-\frac{h}{2})(x-h)$ with $a \neq 0$. Using the point $(0,1)$ we get that $$ a \cdot \left(-\frac{h}{2}\right) \cdot (-h) = 1, $$ so $a = 2/h^2$ (assuming that $h > 0$).