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

Feb 28, 2017

the graph of $y = f \left(x\right)$ 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:

$f \left(x\right) = {x}^{2} - 2 x + 6$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = {\left(x - 1\right)}^{2} - {\left(1\right)}^{2} + 6$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = {\left(x - 1\right)}^{2} - 1 + 6$
$\setminus \setminus \setminus \setminus \setminus \setminus \setminus = {\left(x - 1\right)}^{2} + 5$

And so the graph of $y = f \left(x\right)$ 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 $y = f \left(x\right)$
graph{ x^2-2x+6 [-10, 10, -2, 15] }