Question

How do you find the vertex and axis of symmetry, and then graph the parabola given by: `g(x)= 3x^2 + 12x + 15`?

Answer 1

Vertex: `(-2,3)`
Axis of symmetry: `x=-2`

Explanation:

Given
`color(white)("XXX")g(x)=3x^2+12x+15`

Part 1: The Vertex
Convert into vertex form (`g(x)=m(x-a)^2+b` with vertex at `(a,b)`)

`color(white)("XXX")`Extract the `m`
`color(white)("XXX")g(x)=3(x^2+4x) + 15`

`color(white)("XXX")`Complete the square
`color(white)("XXX")g(x)= 3(x^2+4x+2^2)+15 - 3(2^2)`

`color(white)("XXX")`Re-write as a squared binomial of form `(x-a)^2` and simply
`color(white)("XXX")g(x)=3(x-(-2))^2+3`

This is in vertex form with the vertex at `(-2,3)`

Part 2: The Axis of Symmetry
An parabolic equation in the form:
`color(white)("XXX")y=ax^2+bx+c`
has a vertical axis (i.e. `x=c` for some constant `c`) through the vertex.

Since the given `g(x)` is of this form and the vertex is at `(x,y)=(-2,3)`
the axis of symmetry is
`color(white)("XXX")x=(-2)`