Question

How do I convert the equation `f(x)=x^2-4x+3` to vertex form?

Answer 1

`color(red)( f(x) = (x-2)^2-1)`

Explanation:

The vertex form of a quadratic is given by `y = a(x – h)^2 + k`, where (`h, k`) is the vertex.

The "`a`" in the vertex form is the same "`a`" as in `y = ax^2 + bx + c`.

Your equation is

`f(x) = x^2-4x+3`

We convert to the "vertex form" by completing the square.

Step 1. Move the constant to the other side.

`f(x)-3 = x^2-4x`

Step 2. Square the coefficient of `x` and divide by 4.

`(-4)^2/4 = 16/4 = 4`

Step 3. Add this value to each side

`f(x)-3+4= x^2-4x+4`

Step 4. Combine terms.

`f(x)+1 = x^2-4x+4`

Step 5. Express the right hand side as a square.

`f(x)+1= (x-2)^2`

Step 5. Isolate `f(x)`.

`f(x) = (x-2)^2-1`

The equation is now in vertex form.

`y = a(x – h)^2 + k`, where (`h, k`) is the vertex.

`h = 2` and `k = -1`, so the vertex is at (`2,-1`).