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

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Answer 1 · 2017-02-28T01:10:56

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#).

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] }