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Ethan Tsea
How exactly does the 90 degree rotation
take place? As we can see in the rotated
equations, "y" does not become the answer, but rather a part of the answer, having to be added
to x on the left side, and having x subtracted
from it on the right side. We can see that the
result of these new equations which have been
rotated, equal to their translation of up and
down on the "x" axis. There are several ways
we can write this however - if we look at the
bright blue line, we can see the equation is:
"-y^2+x=2". This is the equivalent of having:
y^2-x = -2. Again, the -2 clearly indicates that
the line has moved 2 units down on the x axis.
Henry
Chan
Vlado Vasile
We can see from the
graph that the changes
which have taken place
are tobe found when
y=x^2 rotates 90 degrees,
this is the main idea, the
main concept that this
exercise has taught,
because the translation
and reflection we have
previously learnt.
Parabolic
Functions
Here, we have a proof of how this works -
when we rotate y=x^2, what we are actually
doing is reflecting it in "y=x", therefore
instantly creating: y^2-x=0 on GeoGebra.
In order to prove this, we have taken 2
points opposite each other on the lines -
(2,4) and (4,2). If we replaced these numbers
we would have y=x^2 as 4=2^2. In order to
keep this true, the rotated line has to follow
the pattern previously mentioned, so that
we would have: 2^2 - 4 = 0, which works
correctly, because both y and x match up
on both equations.
by Ethan,
and
Vlado
Henry
Vlado Vasile
Thank you!
We can see the exact same
thing here - if we had the
two points, (-4,2) and (-2,4),
and put them into the equation,
y and x would be the same on
both lines - this means we have
found a correct method of this
rotation, and from here on, we
can apply other translations
we have learned, such as
moving the lines up and down,
left and right.
Over this whole Parabolic investigation,
there have been several rules we have
encountered from experimenting with
these lines. With a basic line of y=f(x),
we can change it by doing things like:
Basic:
y = f(x) + a - moves up & down
y = f(x+a) - moves left & right
y=-f(x) - reflects on x axis
y=f(-x) - reflects on y axis
In the future, to extend Parabolic patterns
we could look at things like:
y=a f(x) - Stretches graph parallel to y axis
by a factor
y=f(ax) - Stretches graph parallel to x axis
by a factor
These is a very interesting project with
a lot more we would like to explore.