Question

How to describe the relationship between the graph of `f(x) x^2_2x+6` and the graph of `y=x^2`?

Answer 1

the graph of `y=f(x)` is that of `y=x^2` translated one unit to the right (because of `x-1`, and five units up (because of `+5`).

Explanation:

To examine the relationship we need to complete the square of the first function:

`f(x) = x^2-2x+6`
`\ \ \ \ \ \ \ = (x-1)^2 -(1)^2+6`
`\ \ \ \ \ \ \ = (x-1)^2 -1+6`
`\ \ \ \ \ \ \ = (x-1)^2+5`

And so the graph of `y=f(x)` is that of `y=x^2` translated one unit to the right (because of `x-1`, and five units up (because of `+5`).

We can see this graphically:

Graph of `y=x^2`
graph{ x^2 [-10, 10, -2, 15] }

Graph of `y=f(x)`
graph{ x^2-2x+6 [-10, 10, -2, 15] }