Given that $f$ is a quadratic function with minimum $f(x)=f(3)=2$, find the axis, vertex, range, and $x$-intercepts.

Question

College freshman here who sucks at math. Can someone please explain and answer this for me?

Answer 1

HINTS

1. You have a quadratic function, so $f(x) = ax^2+bx+c$. Plug in $f(2)=3$ to get an equation for $a,b$ and $c$.
2. Since $f$ has a minimum, is $a$ positive or negative?
3. Minimum of $f$ should be at $x = -b/2a$. The minimum of $f$ is at $2$, what is a simple equation connecting $a$ and $b$?
4. Combine equations from (1) and (3) to reduce the parabola to one parameter.

Answer 2

A parabola facing up with low point at the origin is $y = ax^2$ for some $a > 0$.

If the low point is at $3$, the equation is $y = a(x-3)^2$.

If the value at $3$ is $2$, the equation is $y = a(x-3)^2+2$.

I don't see how to determine $a$ from the information given.