Answer
$f(x)=-2(x-1)^2+4$
Work Step by Step
Step 1. Assume the quadratic has a standard form $f(x)=a(x-h)^2+k$
Step 2. Given vertex at $(1,4)$, we have $h=1, k=4$, thus $f(x)=a(x-1)^2+4$
Step 3. As $(0,2)$ is on the parabola, we have $2=a(0-1)^2+4$ which gives $a=-2$
Step 4. We conclude that the function is $f(x)=-2(x-1)^2+4$
