Question

How do you find the axis of symmetry, and the maximum or minimum value of the function `y = x^2+ 4x + 3`?

Answer 1

Minimum is `-1`
Axis of symmetry is `x=-2`

Explanation:

This function is a parabola. the parabola has an axis of symmetry that passes through the vertex. In a parabola, the vertex is the highest (if the parabola opens downward) or the lowest point (if the parabola opens up).

The key to solving this problem is finding the vertex.

To do this you could complete the square to rewrite the vertex in vertex form: `y=a(x-h)^2+k` where the vertex is `(h,k)`,

Or... to find the x-coordinate of the vertex you can use `x=-b/(2a)` where `y=ax^2+bx+c`

So in `y=x^2+4x+3` `a=1` and `b=4`.

`x=-4/(2*1)=-2`

Substituting `-2` into your equation, you get `y=(-2)^2+4*(-2)+3=-1`

Your vertex is `(-2,-1)`

So the line of symmetry is a vertical line passing through the vertex: `x=-1`

This parabola opens upward and therefore has a lowest point (minimum and no maximum point) which occurs at the vertex. Since height of the point is controlled by the y-coordinate, the minimum value of the parabola is `y=-1` aka the y-coordinate of the vertex.

graph{y=x^2+4x+3 [-10, 10, -5, 5]}