Question

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Which translation maps the graph of the function f(x) = x² onto the function g(x) = x² − 6x + 6?
Oleft 3 units, down 3 units
Oright 3 units, down 3 units
Oleft 6 units, down 1 unit
Oright 6 units, down 1 unit

Answer 1

• right 3 units, down 3 units

Explanation:

You want the translation that maps f(x) = x² to g(x) = x² -6x +6.

Graph

A graph of the two functions shows g(x) is right 3 units and down 3 units from f(x).

Vertex form

We know the vertex of f(x) = x² is the origin (0, 0). The vertex of g(x) will tell us the translation. Putting that function in vertex form, we have ...

g(x) = x² -6x +6

g(x) = (x² -6x) +6

g(x) = (x² -6x +9) +6 -9 . . . . . add and subtract 9 to complete the square

g(x) = (x -3)² -3

Compare this to ...

y = (x -h)² +k . . . . . . has vertex (h, k)

We see that (h, k) = (3, -3).

g(x) is translated right 3 units and down 3 units.