Graph transformations
Albert
translation of 2 units in the positive direction of the x-axis
translation of 3 units in the negative direction of the y-axis
dilation of factor 2 from the x-axis
reflection in the x-axis
?
As a high school student, may I make the broad statement that one
specifies translations, then dilations, then reflections for quadratics,
cubics, and quartics? Would this also apply to sin and cos?
I may be being vague and if that complicates replies, please tell me
what to specify
Dave L. Renfro
> Is the sequence of transformations from y = x^2 to y = -2(x-2)^2 - 3
>
> translation of 2 units in the positive direction of the x-axis
> translation of 3 units in the negative direction of the y-axis
> dilation of factor 2 from the x-axis reflection in the x-axis
>
> ?
>
> As a high school student, may I make the broad statement that
> one specifies translations, then dilations, then reflections
> for quadratics, cubics, and quartics? Would this also apply
> to sin and cos?
>
> I may be being vague and if that complicates replies, please
> tell me what to specify
The simplest rule is to do the vertical translation last, while
the others can be performed in any order. Of course, technically
speaking, you can do them in any order if you choose the correct
transformations, but by not doing the vertical translation last
you'll have to work harder.
Here are 3 easy ways to go from x^2 to -2(x-2)^2 - 3. There
are 3 more ways I haven't shown, since there are 6 ways in
all that the first 3 steps can be re-ordered (there are
3! = 6 possible ways to arrange 3 things in order).
-------------------------------------------------------------
1. right translate by 2 [replace x with x-2]
x^2 --> (x-2)^2
2. dilate by a factor of 2 [replace f(x) with 2*f(x)]
(x-2)^2 --> 2*(x-2)^2
3. reflect about x-axis [replace f(x) with -f(x)]
2*(x-2)^2 --> -2*(x-2)^2
4. down translate by 3 [replace f(x) with f(x) - 3]
-2*(x-2)^2 --> -2*(x-2)^2 - 3
-------------------------------------------------------------
1. dilate by a factor of 2 [replace f(x) with 2*f(x)]
x^2 --> 2*x^2
2. right translate by 2 [replace x with x-2]
2*x^2 --> 2*(x-2)^2
3. reflect about x-axis [replace f(x) with -f(x)]
2*(x-2)^2 --> -2*(x-2)^2
4. down translate by 3 [replace f(x) with f(x) - 3]
-2*(x-2)^2 --> -2*(x-2)^2 - 3
-------------------------------------------------------------
1. reflect about x-axis [replace f(x) with -f(x)]
x^2 --> -x^2
2. right translate by 2 [replace x with x-2]
-x^2 --> -(x-2)^2
3. dilate by a factor of 2 [replace f(x) with 2*f(x)]
-(x-2)^2 --> -2*(x-2)^2
4. down translate by 3 [replace f(x) with f(x) - 3]
-2*(x-2)^2 --> -2*(x-2)^2 - 3
-------------------------------------------------------------
Here is one of the several hard ways to go from x^2 to -2(x-2)^2 - 3.
1. up translate by 3/2 [replace f(x) with f(x) + 3/2]
x^2 --> x^2 + 3/2
2. right translate by 2 [replace x with x-2]
x^2 + 3/2 --> (x-2)^2 + 3/2
3. dilate by a factor of 2 [replace f(x) with 2*f(x)]
(x-2)^2 + 3/2 --> 2*{(x-2)^2 + 3/2} = 2*(x-2)^2 + 3
4. reflect about x-axis [replace f(x) with -f(x)]
2*(x-2)^2 + 3 --> -{2*(x-2)^2 + 3} = -2*(x-2)^2 - 3
Note that I had to use a vertical shift of 3/2 initially in
order to off-set the later effects of multiplying by 2 and
multiplying by -1. If I had started off with a vertical shift
of -3, then I would have ended up with -2*(x-2)^2 + 6, which
would not be what we wanted to end up with. So you CAN do the
vertical shift first, but if you do so you'll have to "work
backwards" some to make sure you start off with the correct
vertical shift you need at the beginning in order to get what
you want at the end.
Dave L. Renfro