How get the standard form just with x-intersects and vertex coordinates?

Question

I was trying to get the standard form to an equation where:

• The graph intersects the x-axis at $x=−1$ and $x=3$;
• The vertex of the graph lies at $(1,2)$; I can't figure it out even if I got to resolve just using graphs, but instead I want the mathematical path.

Answer 1

Since the roots of the quadratic are at $-1$ and $3$, the factored form should be $y=a(x+1)(x-3)$ for some constant $a$. You have the coordinates of one point on the function, namely the vertex, so plug that into the equation to find out what $a$ is, and then expand to get the standard form.