Question

How do you find the vertex and axis of symmetry, and then graph the parabola given by: `y= x^2 + 6x + 5`?

Answer 1

The axis of symmetry is `x=-3`.
The vertex is `(-3,-4)`

Explanation:

`y=x^2+6x+5` is a quadratic equation which has the form `y=ax^2+bx+c`, in which `a=1, b=6, and c=5`.

Axis of Symmetry

The axis of symmetry is determined by the formula `x=(-b)/(2a)`.

`x=(-6)/(2*1)=-6/2=-3`

The axis of symmetry is `x=-3`

Vertex

The vertex is the point `(x,y)` that is the maximum or minimum of a parabola. Since `a` is positive, the parabola for this graph will open upward and the vertex will be the minimum point.

The `x` value of the vertex is the same as the axis of symmetry. `x=-3`.

To find the `y` value of the vertex, substitute `-3` for `x` in the equation `y=x^2+6x+5`.

`y=(-3)^2+6(-3)+5=`

`y=9-18+5=-4`

The vertex is `(-3,-4)`.

Determine several points on both sides of the axis of symmetry by substituting values for `x` into the equation and solving for `y`.

`y=x^2+6x+5`

`x=-6,` `y=5`
`x=-5,` `y=0`
`x=-4,` `y=-3`
`x=-3,` `y=-4` (vertex)
`x=-2,` `y=-3`
`x=-1,` `y=0`
`x=0,` `y=5`

Plot the points and sketch an upward opening parabola with the curve at the vertex. Do not connect the dots.

graph{y=x^2+6x+5 [-16.02, 16, -8.01, 8.01]}