I think he is referring case of k = -2 which I missed.
To shift a graph k units to the left the transformation is f(x+k) where k is positive.
Yes, I am.
And
@Dashdorm24, you can check each is valid by doing the transformation and seeing that the resulting
has
as a factor.
For example, taking
(which is the same as
, I'm just using
as a dummy variable so that the transformation is clearer), and performing the transformation
, as suggested by
@cossine, gives:
and it has a single root at the origin and a double root at
, just as would be expected from a shift of one unit to the right.
The second transformation,
, which
@cossine correctly identified from my hint, should produce a shift of 2 units to the left, moving the double root to the origin. Checking:
and it has a single root at
and a double root at the origin, just as would be expected from a shift of two units to the left.
Noting that our given polynomial
has
, we can also predict that a shift downwards by 4 should give a transformed polynomial with a root at the origin. Check:
and it has a single root at
and a double root at the origin, just as would be expected from a shift of four units to the down. (The point
lies on the original polynomial, so a shift down of four should produce a root of the new polynomial at
.)
These techniques can be combined, along with dilations, to produce other polynomials as required.