Question

Find the vertex and axis of symmetry of this: `y = -3 (x + 4)^2 +2`?

Answer 1

The vertex is the point of coordinates `(-4,2)`.
The axis of symmetry is the (vertical) line of equation `x=-4`.

Explanation:

In the equation `y=−3(x+4)^2+2` the variable `y` is a quadratic function of `x`. So the graph of the function in the `(x,y)` plane is a parabola.

The equation of a general parabola can be expressed in two ways:

• Standard Form: `y=ax^2+bx^2+c` where `a,b,c in RR` are arbitrary coefficients.
• Vertex Form: `y=a(x-h)^2+k` where `a in RR` is an arbitrary coefficient and `(h,k) in RR^2` are the (arbitrary) coordinates of the vertex.

In this case, the equation is written in the Vertex Form. This form is especially useful to find the vertex and the axis of symmetry.

It's easy to find the vertex: since we have the Vertex Form, we can detect the values of `-h` ad `k`: `-h=4` and `k=2`. Hence `(h,k)=(-4,2)` is the vertex of this parabola.

The axis of symmetry comes naturally from the information on the vertex, because the axis of symmetry intersects the parabola always at the vertex `(h,k)`. Since the equation is expressed as a function of `x`, the axis of symmetry has to be vertical. So its equation is `x=h` i.e. `x=-4`.