Question

How do you tell whether the graph opens up or down, find the vertex and the axis of symmetry of `y=3(x+4)^2-2`?

Answer 1

See below.

Explanation:

When a quadratic is arranged in the form `a(x - h)^2 + k`
`k` is the minimum or maximum of the function.
`h` is the axis of symmetry.
From example:

vertex is `( -4 , -2)`
axis of symmetry is `- 4`

If `a` is negative, then the parabola is inverted i.e. `nnn`

If `a` is positive, then the parabola `uuu`

Answer 2

As the `x^2` term is positive the graph opens upwards

Vertex `->(x,y)=(-4,-2)`

So axis of symmetry is `x=-4`

Explanation:

If you expand the brackets and multiply by the three the first term is

`+3x^2`

As this is positive the graph is of general shape `uu`

Suppose it had been negative then in that case the graph would be of form `nn`

The given equation format is of type 'completing the square' also known as 'vertex form'.

`y=3(x+color(red)(4))^2color(blue)(-2)`

`x_("vertex")=(-1)xxcolor(red)(4) = -4`

`y_("vertex")=color(blue)(-2)`

Vertex `->(x,y)=(-4,-2)`