Question

What is the axis of symmetry and vertex for the graph `y = x^2 - 4x + 1`?

Answer 1

`x = 2` is the line of symmetry.

`(2,-3)` is the vertex.

Explanation:

Find the axis of symmetry first using `x = (-b)/(2a)`

`y = x^2-4x+1`

`x= (-(-4))/(2(a)) = 4/2 = 2`

The vertex lies on the line of symmetry, so we know `x = 2`
Use the value of `x` to find `y`

`y = (2)^2 -4(2) +1`

`y = 4-8+1 = -3`

The vertex is at `(2,-3)`

You can also use the method of completing the square to write the equation in vertex form: `y= a(x+b)^2 +c`

`y = x^2 -4x color(blue)(+4-4) +1" "[color(blue)(+(b/2)^2-(b/2)^2)]`

`y = (x-2)^2 -3`

The vertex is at `(-b,c) = (2,-3)`