Question

What is the axis of symmetry and vertex for the graph `x^2 + 4x - 6y + 10 = 0`?

Answer 1

Vertex: `(-2,1)`
Axis of symmetry: `x=-2`

Explanation:

Given
`color(white)("XXX")x^2+4x-6y+10=0`

with minor rearranging we can write this in standard quadratic form as:
`color(white)("XXX")y=1/6x^2+2/3x+5/3`
or (as will be more convenient later):
`color(white)("XXX")y=1/6(x^2+4x)+5/3`
which is a parabola with a vertical (parallel to the y-axis) axis of symmetry.

We can convert this second version into vertex form by completing the square:
`color(white)("XXX")y=1/6(x^2+4xcolor(red)(+4))+5/3color(red)(-(1/6) * (4))`

`color(white)("XXX")y=1/6(x+color(blue)(2))^2+color(green)(1)`
or
`color(white)("XXX")y=1/6(x-(color(blue)(-2)))^2+color(green)(1)`

Remember that the general vertex form is
`color(white)("XXX")y=color(brown)(m)(x-color(blue)(a))^2+color(green)(b)` with vertex at `(color(blue)(a),color(green)(b))`

So the vertex of this parabola is at `(color(blue)(-2),color(green)(1))`

The axis of symmetry goes through the vertex and as already noted is parallel to the y-axis;
so it's equation is
`color(white)("XXX")x=-2`

Here's a graph of the original equation for verification purposes:
graph{x^2+4x-6y+10=0 [-7.035, 2.83, -0.65, 4.28]}

Answer 2

The axis of symmetry of the given curve

(Parabola) is the line `: x+2=0`, and, the vertex `(-2,1)`.

Explanation:

We rewrite the eqn. as, `x^2+4x=6y-10`, &, now, complete the

square on the L.H.S., so that,

`x^2+4x+4=6y-10+4, "i.e.," (x+2)^2=6(y-1)............(1)`.

Shifting the Origin to the pt. `(-2,1)`, suppose that the pt. `(x,y)`

becomes `(X,Y)`.

The conversions, as known from the Co-ordinate Geometry, are,

`x=X-2, y=Y+1`, or, `x+2=X, y-1=Y.................(2)`.

Hence, `(1)` becomes, `X^2=6Y`,

which, represents a Parabola, having, the new Y-axis

[eqn. `X=0`] as its the axis of symmetry, and, the Vertex

`(0,0)` in `(X,Y)` system.

By `(2)`, we conclude that, the axis of symmetry of the given curve

(Parabola) is the line `: x+2=0`, and, the vertex `(-2,1)`.

Enjoy Maths.!