Question

Which translation maps the graph of the function f(x) = x2 onto the function g(x) = x2 − 6x + 6?

Answer 1

Answer:

As given

f(x) = x² and g(x) = x² - 6 x +6

Preimage = x² and Image = x² - 6 x + 6

Both f(x) and g(x) are quadratic function i.e function having highest degree 2 is a parabola.

Now , the meaning of word translation is moving the curve or function to different position i.e in the same coordinate system but position is different.

We will find the solution by options.

1.y= f(x) = x²

Translation by left 3 units, down 3 units.

Let consider point (x,y) on the parabola.By translation left 3 units, down 3 units we moved to (p,q).

p= x - 3, q = y-3→x= p+ 3, ∧ y = q+3

Putting in the equation , y = x²

→ q + 3 = (p+3)²= p²+ 6 p + 9→f(x)=q = p² + 6 p +6,≠g(x) which is untrue.

Option 2.

y = x²

Translation by right 3 units, down 3 units gives

→f(x)=y +3 = (x -3)²=x² - 6 x +9

f(x)=y = x²-6 x + 9 -3

 = x² - 6 x + 6=g(x)

Don't need to look at other options.

Option 2 i.e Translation by right 3 units, down 3 units is correct option.

Answer 2

A parent function f(x) can be translated to the right by h and up by k using the the following rule:

G(x)=f(x-h)+k, where (h,k) is the translation vector.

f(x)=x^2

G(x)=f(x-h)+k=x^2-6x+9 -3 = (x-3)^2 -3

so h=3, k=-3.

Recall the rule: h>0 => right,  k>0 => up.