Question

How to find the quadratic equation when given the vertex and y intercept?

Answer 1

Suppose you are given a quadratic equation with vertex (h, k) and y-intercept, c.

The vertex form of a quadratic equation is given by:


`y=a(x-h)^2+k=ax^2-2ahx+ah^2+k`

y-intercept is the value of y when x is 0.

Thus, given a y-intercept, c, we have:


`c=ah^2+k \\ \\ \Rightarrow ah^2=c-k \\ \\ \Rightarrow a= (c-k)/(h^2)`

Therefore, given a vertex (h, k) and y-intercept, c, the quadratic equation is given by


`y=\left((c-k)/(h^2)\right)x^2-2\left((c-k)/(h^2)\right)hx+\left((c-k)/(h^2)\right)h^2+k \\ \\ =\left((c-k)/(h^2)\right)x^2-2\left((c-k)/(h)\right)x+c-k+k \\ \\ =\bold{\left((c-k)/(h^2)\right)x^2-2\left((c-k)/(h)\right)x+c}`