Question

What is the axis of symmetry and vertex for the graph `y=x^2-4x+5`?

Answer 1

Axis of symmetry: `x=2`
Vertex: `{2,1}`

Explanation:

Let's transform this function into a full square form:
`y=x^2-4x+5=x^2-4x+4+1=(x-2)^2+1`

Using this, we can transform the graph of `y=x^2` into `y=(x-2)^2+1` by performing the following steps:

Step 1
From `y=x^2` to `y=(x-2)^2`
This transformation shifts the graph of `y=x^2` ( with axis of symmetry at `x=0` and vertex at `{0,0}` ) to the right by 2 units.
Axis of symmetry also will be shifted by 2 units and now will be at `x=2`. The new vertex position is `{2,0}`.

Step 2
From `y=(x-2)^2` to `y=(x-2)^2+1`
This transformation shifts the graph of `y=(x-2)^2` up by 1 unit.
Axis of symmetry, as a vertical line, would be transformed into itself.
The vertex will move up by 1 unit and be at `{2,1}`.