# How to find the quadratic function given the vertex (2,-4) and the point (1,-1)?

Jun 4, 2015

Since the vertex is at $\left(2 , - 4\right)$ we can deduce

$f \left(x\right) = a {\left(x - 2\right)}^{2} - 4$

for some constant $a$

Then from the point $\left(1 , - 1\right)$ we get:

$- 1 = f \left(1\right) = a {\left(1 - 2\right)}^{2} - 4 = a - 4$

Add $4$ to both ends to get

$a = 3$

So:

$f \left(x\right) = 3 {\left(x - 2\right)}^{2} - 4 = 3 {x}^{2} - 12 x + 8$