- [Tutor] So this is a screenshot of Desmos,
it's an online graphing calculator,
what we're gonna do is use it to understand
how we can go about scaling functions
and I encourage you to go to Desmos and try it on your own
either during this video or after.
So let's start with a nice, interesting function,
let's say f of x is equal to the absolute value of x,
so that's pretty straightforward.
Now let's try to create a scaled version of f of x,
so we could say g of x is equal to,
well, I'll start with just absolute value of x,
so it's the same as f of x,
so we'll just trace the g of x right on top of f,
but now let's multiply it by sum constant,
let's multiply it by two.
So notice the difference between g of x and f of x
and you can see that g of x is just two times f of x,
in fact we can write it this way,
we can write g of x is equal to two times f of x,
we get to the exact same place,
but you can see that as our x increases,
g of x increases twice as fast, at least for positive xs
on the right-hand side and actually as x decreases,
g of x also increases twice as fast,
so is that just a coincidence that we have a two here
and it increased twice as fast?
Well, let's put a three here,
well now it looks like it's increasing three times as fast
and it does that in both directions.
Now what if we were to put a 0.5 here, 0.5?
Well now it looks like it's increasing half as fast
and that makes sense, because we are just multiplying,
we are scaling how much our f of x is.
So before when x equals one, we got to one,
but now when x equals one, we only get to one half,
before when x equals five, we got to five,
now when we get to x equals five, we only get to 2.5,
so we're increasing half as fast,
or we have half the slope.
Now an interesting question
to think about is what would happen
if instead of it just being an absolute value of x,
let's say we were to have a non-zero y intercept,
so let's say, I don't know, plus six,
so notice then when we change this constant out front,
it not only changes the slope,
but it changes the y intercept,
because we're multiplying this entire expression by 0.5,
so if you multiply it by one,
we're back to where we got before
and now if we multiply it by two,
this should increase the y intercept,
'cause remember we're multiplying both of these terms
by two and we see that, it not only doubles the slope,
but it also increases the y intercept.
If we go to 0.5,
not only did it decrease the slope by a factor of one half,
or I guess you could say multiple the slope by one half,
but it also made our y intercept
be half of what it was before
and we can see this more generally
if we just put a general constant here
and we can add a slider
and actually let me make the constant go from zero to 10
with a step of, I don't know, 0.05,
that's just how much does it increase
every time you change the slider
and notice when we increase our constant,
not only we're getting narrower,
'cause the magnitude of the slope is being scaled,
but our y intercept increases and then as k decreases,
our y intercept is being scaled down
and our slope is being scaled down.
Now that's one way that we could go about scaling,
but what if instead of multiplying
our entire function by sum constant,
we instead just replace the x with a constant times x,
so instead of k times f of x,
what if we did it f of k times x?
Another way to think about it is g of x
is now equal to the absolute value of kx plus six,
what do you think is going to happen?
Pause this video and think about it.
Well now when we increase k,
notice it has no impact on our y intercept,
because it's not scaling the y intercept,
but it does have an impact on slope,
when k goes from one to two,
once again we are now increasing twice as fast
and then when k goes from one to one half,
we're now increasing half as fast.
Now this is with an absolute value function,
what if we did it with a different type of function,
let's say we did it with a quadratic?
So two minus x squared,
let me scroll down a little bit
and so you can see when k equals one, these are the same
and now if we increase our k,
let's say we increase our k to two,
notice our parabola is in this case decreasing
as we get further and further from zero
at a faster and faster rate,
that's because what you would have seen at x equals two,
you're now seeing at x equals one,
because you are multiplying two times that
and so then if we go between zero and one,
notice on either side of zero,
our parabola is decreasing at a lower rate,
it's a changing rate, but it's a lower changing rate,
I guess you could put it that way
and we could also try just to see
what happens with our parabola here,
if instead of doing kx, we once again put the k out front,
what is that going to do?
And notice that is changing not only how fast
the curve changes at different points,
but it's now also changing the y intercept,
because we are now scaling that y intercept.
So I'll leave you there,
this is just the beginning of thinking about scaling,
I really want you to build an intuitive sense
of what is going on here
and really think about mathematically why it makes sense
and go on to Desmos and play around with it yourself
and also try other types of functions and see what happens.