Question

What is the axis of symmetry for f(x) = −3x2 + 12x − 6?

Answer 1

The axis of symmetry can be found by finding the average of the zeros, a derivation from the conservation of energy :P, or by finding the point when the velocity is equal to zero.

df/dx=-6x+12 so df/dx, velocity, equals zero when:

-6x+12=0

6x=12

x=2  so the axis of symmetry is the vertical line x=2

....

average of zeros...

3x^2-12x+6=0

x^2-4x+2=0

x^2-4x=-2

x^2-4x+4=2

(x-2)^2=2

x-2=±√2

x=2±√2  so the average of the zeros is obviously 2.

....

conservation of energy

vf-vi=at  When vf=0, this is the maximum value for f(x)...

-vi=at, vi=b and a(acceleration)=2a(from quadratic) and t=x

-b=2ax

x=-b/(2a)  in this case

x=-12/(2(-3))

x=-12/-6

x=2

 

Answer 2

Answer:  The axis of symmetry is  $x=2.$

Step-by-step explanation:  The given function is

$f(x)=-3x^2+12x-6~~~~~~~~~~~~~~~~(i)$

We know that for the function $f(x)=a(x-h)^2+k,$ the axis of symmetry is given by

$x=h.$

From equation (i), we have

$f(x)=-3x^2+12x-6\\\\\Rightarrow f(x)=-3(x^2-4x)-6\\\\\Rightarrow f(x)=-3(x^2-4x+4)+12-6\\\\\Rightarrow f(x)=-3(x-2)^2-6.$

Therefore, the axis of symmetry for the given function is

$x=2.$

Also, the graph of the function is shown in the attached figure. It is a parabola with vertex (2, 6) and axis of symmetry x = 2.