Question

Identify the vertex, axis of symmetry, and min/Max value:

f(x)=3x^2-54x+241

Show work

Answer 1

Well, the vertex is the turning point right? Let's start there. First differentiate f(x)

Answer 2

What do you get when you differentiate f(x)?

Answer 3

Alright, now that is a quadratic equation, so the equation is a parabola, and since the "ax^2" term has a positive 'a' value, the graph points upwards. We know this much from the equation. Now, usually, we factorize the equation into terms like (ax+b)(cx+d) if we can, and that usually helps, but in this case, that is impossible since 241 is a prime number.
alright, so, we can try the VERTEX FORM of a parabola, which is, of the form a(x-h)^2 +k, when the vertex will be (h,k)..
If you try to convert it into that form, you get the equation, f(x) = 3(x-9)^2 -2 ... So, the Vertex is (9,-2), and like I said before, since the 'a' term is positive, this will be the parabola's one and only minumum.
Also, the axis equation is of the form x= -b/2a, so in this case, b = -54, a = 3, therefore ,the axis equation is x = 9.