We're going to be looking at transformations
of functions today and I want you to just
watch this video so you can kind of visually
see what each transformation does you're not
going to have these in your notes packet you're
just going to watch what I'm doing so you
can see each transformation one at a time
separately but these ones we're using absolute
value functions and the reason for this is
because I think it's the easiest to see visually
what's going on when we use absolute value
it's a v-shaped graph it's really easy to
see how these graphs are shifting and changing
and that's why I've chosen this one specifically
I'm going to graph each transformation in
different colors so you can see what it looks
like we're going to start with our parent
function and red for this first one here I'm
going to zoom into it and this is what our
parent function looks like
it's taking the absolute value of each number
which we talked about the absolute value measures
distance so I want the positive version of
whatever number I'm dealing with so in this
case when I obviously that's one but when
I took the absolute value of negative one
that one was also one that's about right here
when I took the absolute value of -1 it was
also one that's that point here that's my
parent function that means every absolute
value starts like this and then we transform
it to make all of our different possible absolute
value graphs and so the next one the one that
I'm going to Grafton blue this one says now
what if I took that parent function y equals
absolute value of x but then I added three
so we're going to take that same idea so if
I started with the absolute value 0.0 and
then I add three to it that's going to now
put that vertex point that lowest point up
here then if I take the absolute value of
one that's still one and then I add three
to it that's going to make it for which is
this point right there if I take the absolute
value of -1 so this point over here that's
positive one and then I add my 3 it's going
to go right here and that shape is just going
to continue out like it did before and then
I'm going to connect my points together so
you can see
hey so now we can look at our parent function
what this graph started off as and we can
see what's happened in this new one and so
if you look at the very top of this page under
the instructions you have this right here
cuz that's y equals a absolute value x minus
H + K and obviously when you look at these
there's a lot less letters a lot less variables
that were using and the reason for this is
the letters a and H and K are actually telling
us different Transformations that are happening
I'm going to go ahead and grab this last one
here this green one so you can see the other
transformation but you can see that we're
dealing with this part of that General function
of the top we're dealing with the plus que
parte and seeing what this specifically does
so that we can determine a rule that will
help us in the future so now I'm going to
change my color to Green in the same ideas
for this one's -5 so if I had to take the
absolute value of 080 - 5 gives me -5 which
puts up first point all the way down here
now if I take the absolute value of one it's
one but then I subtract 5 it's going to be
negative for which is actually right there
absolute value of negative one is positive
1 and 7. Case when I subtract my five it's
also going to be negative for so again you
can kind of see that V shape forming I'm going
to drop my lines to connect them together
so you can see this transformation as well
and it looks like that zoom out so you can
see all three at once we looking at the similarities
and differences between these two and we're
going to write a rule for how changing K so
again in this part of my general function
so what happens when we change queso if we
look at the blue one when we added something
so when we add a value it actually moved that
graph the red graph and actually shift so
I'm going to put that shift up so when we
add a k value to the end of our function we're
going to be moving that graph up and then
we subtract it so that's are green when we
subtracted 5 it actually moved our graph down
so subtracting 
value ship sit down so at the very end of
our general function here we're looking at
how is this affecting our grass how is it
changing how's the transforming and so are
+ K on the end if it's + K2 moving up if it's
minus K is going to be moving it down so that's
our first transformation will just find these
in a minute that's actually what your notes
are going to be but now you can just kind
of visually see how much going on these nuts
will be posted into our schoology page at
the end of the week with all of our notes
for this entire unit our next one now he's
changed a little bit instead of it being outside
of the absolute value it's actually inside
of the absolute value with RX variable and
so you can look up at the top again this is
H so we're changing H What's Happening Here
into our first one we're going to graph is
just our parent function again some to go
back to my red value and I'm just going to
re graph that parent function that I had at
the beginning last time so we can see how
this is changing our graph and then we're
going to do our blue graph first and that
one says why is equal to the absolute value
of x + 2 so I'm like before we took the absolute
value then we added something to it this time
it's inside absolute value so you have to
add it first and then take the absolute value
and so when I have zero on the inside zero
plus 2 is 2 and the absolute value of that
1 + 2 is 2 1 + 2 is 3 the absolute value of
that is 3 2 + 2 is 4 and the absolute value
of that is for I can kind of see my shape
but it seems like they're all just going up
something to go to the other side to the negative
one negative one plus two is one the absolute
value of that is one negative 2 plus 2 is
0 and the absolute value of that is 0 so now
I'm kind of going down -3 + 2 is -1 but the
absolute value of that is positive once I
found my Turning Point here will do one more
so you can kind of see the shape so we're
at -4 + -4 + 2 + -2 and then you take the
absolute value of that witches are positive
to so now you can see are v-shaped we've created
here it was a little bit trickier because
you couldn't just use the same values as the
parent function cuz it's not lining up quite
as well you can see another shift has occurred
here and so now we're going to look at the
green one so we can see what the difference
is when we subtract this one says - 4K so
again I'm going to start at 0 so if I take
0 -4 that's negative for but the absolute
value that is positive for so we're all the
way up here I'm going to do positive one again
so 1 - 4 is -3 but again we're taking an absolute
value sets me that point right there positive
3 to -4 is -2 but again absolute value or
take an absolute value so it's positive to
3 - 4 is -1 the absolute value is positive
1 4 -4 0 with touchdown absolute value that
is 0 5 - 4 is one absolute value of 1 6 - 4
is to absolute value still too and now we
can see her shape going up the other way I'm
going to connect these together add my arrows
and now I can see what's happening for this
one here hey so again we're dealing with the
number that's inside of our absolute values
which again is this value H here at the top
so for this one whenever we are adding a value
and actually went to the left so we think
about left and right we're thinking about
the negative values are to the left the positive
values are to the right but it's actually
the opposite when's Lil bit more notes today
I'm just going to write adding value goes
to the left 
adding value shifts left when we subtract
the value it's shifted to the right and you
can see that clearly in our pictures here
of our parent functions versus are left and
right so now you do find one now we're shifting
went way too far now we're talkin about this
number at the very beginning right in front
of our absolute value is going to be our next
two Transformations were dealing with that
this one specifically wants to know what's
different about it being negative or positive
so obviously this one is our parent function
that's our positive version to my go ahead
and grab that parent function again same as
before or parent function isn't changing American
access points together answer for this one
or negative value is outside of absolute value
bars so wants to know after I take the absolute
value we're putting a negative on it so zero
is obviously not changing the absolute value
of zero zero negative zero is not a thing
that's till that point right there but what
about if I meant positive one so this is what
it would look like I take the absolute value
of 1 and that's just one but that negative
outside so it's actually going to be planted
at negative one because that negative gets
applied after I take the absolute value if
I'm on the other side and I'm trying to take
negative one remember there's still a negative
on the outside when I take the absolute value
of negative one is positive one but I need
to bring down that negative sign that was
outside of my absolute value bars so it actually
looks like that so you can kind of see the
direction it's going to go I'm going like
this so you might already know the name for
this hopefully you guys have seen this one
before but this is actually a reflection some
people like to say it flipped it across the
x-axis and that's just a different way of
saying we reflected across the x-axis negative
causes a reflection
across the x axis so if I had a mirror sitting
there on the x-axis you would see that mirror
image flipped over and that's all this one
was doing our last one is our tricky one so
this one I'm going to drop my parent function
same as before just like that and now we have
something different because there are numbers
in front this means it's being multiplied
so my value that's in blue so I have four
times the absolute value of x so damn sleepy
0 times 4 is still zero semi vertex is sticking
here on the same place now if I take the absolute
value of one it's going to be one but then
I'm multiplying it by force that's going to
be positive for which is all the way up here
now if I take the absolute value of -1 that's
also positive one and then we multiply are
for inside of (that's going to be at 4 Johnny
other side we can do that one two three and
four which is all the way up there you can
already see your shape I'm going to do one
more so you can see this cuz we can fit it
to buy take the absolute value of 2 that's
2 * 4 is 8 all the way up there absolute value
of -2 is 2 * 4 is also 8 I'm going to connect
those together so you can see what happens
are in there so you can see the difference
between the red and the blue graph obviously
the Bluegrass is a lot thinner a lot narrower
than the red grass was and then we're going
to do our green grass this time or multiplying
it by 1/4 0 stays the same I'm going to give
you numbers that are nice and even and not
crazy decimals so we're actually going to
start with an x value of 4 to the absolute
value of 4 is 4 and then I went to times it
by 1/4 which is actually just one so I'm going
to go all the way over to where X is 4 and
we're going to put a. They're at positive
one now the same woodwork if it was a negative
for so negative for is positive for when you
take the absolute value but then it's going
to give us a positive one once we * 1/4 I'm
going to do 8 the same way so that's a value
of 8 is 8 * 1/4 is actually just two and the
same is true about -8 and so the opposite
of what happened with the blue is happening
now with the green so now my graph is actually
a lot wider it almost looks like it's a book
that's closing up so the bigger the number
the thinner it is the narrower it is the smaller
the number The Wider it's going to be okay
and so the reason for this is when that number
is really big so in the case for that was
making my y values grow super quickly because
I was multiplying by 4 it's making those values
grow much quicker than the red one and way
way way way quicker than the green one the
green one is making those grow a lot slower
hey something to put that when it's bigger
than one so when is bigger than one this is
going to make it narrower dinner skinnier
and then one day is less so when a is less
than one it's going to be bigger than 0 so
some form of fraction or decimal this is going
to make it wider so now we're going to look
at these last couple of questions pretty quickly
it says how will the graph y equals x absolute
value of x - 6 or in a look back at where
that looks like that looks like its first
one right here in Surrey said when we subtract
it's going to shift it down hey so this one
right here 
draw are is it right here is affecting the
K on the end so it's affecting this value
and so because it's subtraction that means
this is going to shift it down and it's actually
going to go down by that specific amount so
6 units or 6 spaces on our coordinate plane
phone number to go ahead and write the general
form here so just how will the graph y equals
absolute value of x + 1 compared to the graph
of y equals just absolute value of x so that
one is affecting this part inside and said
it was our second edition so we said when
we added value we're actually moving it to
the left that's like our blue shape that we
made and so in this case when I'm adding one
I'm actually going to shift left one 
the next one has how will the graph of y equals
3 x absolute value of x - 10 compared to our
parent function again I'm going to write our
general equation off to the side so we can
see what pieces are being affected specifically
queso in this case we have a number in the
front we have a 3 out here and we have a number
on the inside there's actually two things
going on and so in this case that number is
bigger than one so we talked about how that
makes it narrower so it's going to be narrower
and we have the number again just like the
question about number 6 that's inside and
we're subtracting this time so when we subtract
were actually going to the right so we shift
right and this one is 10 Pacific Lee k&r last
song before we take some notes on this we
have y equals -5 x - 2 and then + 10 on the
outside this one we have a being affected
so we got a & H n k in the front we have a
-5 to the negative is actually making that
reflection talk about that flipping it over
such a reflection and then we just look at
this five by itself five is bigger than ones
this one's going to be narrower then we have
the -2 inside of (and so in this case we're
going to the right two and then on the end
we have questions or actually shifting up
10 you can kind of see how he can have multiple
Transformations you can have one you can have
two you can have three you got four or you
can have none but you can't have more than
that so you can't shift left and right you
can't shift up and down you can't have something
to become wider and narrower but you can have
lots of combinations of either or of those
ones to take some notes now
