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# Intro to graphs of absolute value functions

Video transcript

Let's think a little bit about
the graphs of absolute value functions. And I've defined one
right over here. f of x is equal to negative 3
times the absolute value of x minus 1 plus 9. And then we've
constrained its domain. This is for x
being-- or negative 4 is less than or equal
to x, which is less than or equal to 5. And I encourage you
to pause this video and try to graph
this on your own. And clearly you could graph
this by just trying out different x-values and plotting
the corresponding x and f of x values, but really
think about the structure of this expression
to think about where does it hit a
maximum or minimum point? And where would the
vertex of it actually be? So let's look at this
just holistically before we even try some points. So we have an absolute value. This absolute value
here is clearly going to always be
non-negative, and then we're multiplying it by
a negative number. So this entire thing,
you have a negative times a non-negative, so this
whole thing is always going to be non-positive. The reason why I say
non-positive instead of negative is that
because it could be 0. So this whole part
of the expression is either going
to be 0, or it is going to be a negative number. Now, if this is
always non-positive, what is the maximum
value of this function? Well, the maximum
value happens when the absolute value
is equal to 0. When the absolute value part of
the expression is equal to 0, then this part is 0, and the
function hits its maximum value at f of x is equal to 9. So when does our absolute
value part of this expression equal 0? Well, let's think about it. When does the absolute
value of x minus 1 is going to be equal to 0
if x minus 1 is equal to 0. If x minus 1 is equal to 0--
or if you add 1 to both sides-- you get x is equal to 1. So when x equals 1, we
hit our maximum point. This thing becomes
0, which makes this whole thing equal to 0. And so we get the
point 1 comma 9. So let's graph that point. And actually, let me
do it in a color that's easier to see on a
white background. 1 comma 9 is the maximum
point right over here. And you could also
view this as the vertex of this absolute value function. So 1 comma 9 is that
point right over there. And so we can already guess
the general shape of this. An absolute value
function either looks something like that, or
it looks something like that. It would look like this if the
coefficient right over here-- or i guess maybe I shouldn't
call it a coefficient. If the thing multiplying the
absolute value was positive, then would it be upward
opening like that. But since it is negative, it is
going to be downward opening. We hit a maximum point
at the vertex in the case we're just talking about for
the reasons we just discussed. But then what happens there? What happens to the
right of that vertex, and what happens to the
left of this vertex? We don't have to do it this way. Once again, you could just
try out points if you like, but I like to really analyze
what's going on here. So let's break up this function
definition into two pieces. So let's say that f
of x is equal to-- and we're going to break
it up into two pieces, one piece to the right
of the vertex and one piece to the left of the vertex. So to the right of
the vertex, that's going to be 1 is less than or
equal to x, which is less than or equal to 5. And to the left
of the vertex-- so this is to the
right of the vertex. To the left of the vertex,
that's going to be-- let me do this in a different color. That's going to be negative
4 is less than or equal to x, and I'll just say less than 1. So just to be clear,
our whole domain was between negative 4 and 5. Between negative 4 and
5 was our entire domain. Now, I'm breaking it up
to the right of the vertex and the left of the vertex. So this right over here is
to the right of the vertex, and this right over here is
to the left of the vertex. So to the right of
the vertex, for x being greater than or equal to
1, this absolute value-- or one way to think about it, when x
is greater than or equal to 1, x minus 1 is going
to be positive, and the absolute value
of a positive thing is just going to
be that same value. So for x is greater
than or equal to 1, our function is going to be
negative 3 times x minus 1 plus 9. Or if you wanted to
distribute the negative 3, this would be the same thing. Actually, let me write
it over here so that we can write the
simplified version. So it would be-- let me
give myself some space. So for x is greater
than or equal to 1, we don't have to think
about the absolute value. So it would be negative
3 times x minus 1 plus 9, which is negative
3x plus 3 plus 9, or negative 3x plus 12. So it's going to be
negative 3x plus 12 when x is greater
than or equal to 1. And the really interesting
thing, the slope here is negative. We have a slope of negative 3. So starting from this point, if
we have a slope of negative 3, that means every time we move
1, increase in the x-direction, we're going to decrease
by 3 in the y-direction, increase by x,
decrease by y by 3. So it's a slope of negative 3. So it will look like that. And let me actually
do it in purple so that you can actually see it. That's not purple. There you go. That's purple. All right. So it would look like that. And actually let me
just make it clear. This is what I'm talking
about right over here. That piece of the function
is going to look like that. Now, what about when
x is less than 1? Well, when x is less
than 1, x minus 1 is going to be negative,
and so the absolute value is going to be the negative
of that actual function. So when x is less than
1-- let me write this. The absolute value
of x minus 1 is equal to the negative of x
minus 1, which is the same thing as 1 minus x. So our function is going to
be negative 3 times 1 minus x plus 9, which is the
same thing as-- let's see, if you distribute the
negative 3, it's negative 3 plus 3x plus 9, which
is equal to 3x plus 6. So the important
thing to realize here is that our slope is positive 3. And you could even say that we
have a y-intercept here at 6, so it would be this
point right over here, and the line would
look like this. When your slope is positive
3, you increase x by 1, you increase y by 3. Or another way of thinking about
it, if you decrease x by 1, then you decrease y by 3. So it would look like that. And we are done. We have graphed this
absolute value equation. Now, the whole
reason why I did this is just so we learn it
a little bit deeper. But there would have been
a faster way of doing this. You could have just
said, hey, this is going to hit a maximum
point when x is equal to 1 because this non-positive
thing is going to hit 0. So you could have said
1 comma 9 is the vertex. And then you could have just
tried out the two endpoints, knowing that between the
endpoints and the vertex, you're essentially just
dealing with a line. So you could have just evaluated
f of negative 4 and f of 5. You could have figured out these
endpoints and drawn the lines. But now we understand it
deeper, and that's always a little bit more exciting.