Question

The safety inspector notes that Ray also needs to plan for a vertical ladder through the center of the coaster's parabolic shape for access to the coaster to perform safety repairs. Find the vertex and the equation for the axis of symmetry of the parabola, showing your work, so Ray can include it in his coaster plan.
f(x)= 3x^2+9x+12

Answer 1

Answer with explanation:

Equation of the Parabola is

$f(x)= y=3x^2+9x+12\\\\y=3[x^2+3 x+4]\\\\y=3[(x+\frac{3}{2})^2+4-(\frac{3}{2})^2]\\\\y=3[(x+\frac{3}{2})^2+(\frac{\sqrt 7}{2})^2]\\\\y-\frac{21}{4}=3[(x+\frac{3}{2})^2]$

Vertex of the parabola can be obtained by

   $x+\frac{3}{2}=0\\\\ x=\frac{-3}{2}\\\\ y-\frac{21}{4}=0\\\\y=\frac{21}{4}\\\\ Vertex(\frac{-3}{2},\frac{21}{4})$

Axis is that line of parabola which divides the parabola into two equal halves.

$x+\frac{3}{2}=0\\\\x=-\frac{3}{2}$  

Answer 2

You can rearrange f(x) as:

f(x) = 3(x^2 + 3x) + 12 = 3(x+1.5)^2 +5.25

So you can see the vertex is at (0,5.25) when x = -1.5

The axis of symmetry then lies in x = -1.5