We're asked to graph the function
y = 2sin(-x) on the interval
the closed interval so it includes the endpoints
-2π to 2π
So to do this
I'm going to graph the function y = sin(x)
and then think about how it's changed by the 2
and the negative in front of the x over here
So let's look at the sine of x first
So let me draw our x-axis
let me draw the y-axis
pretty straight forward
and we care about it
between -2π and 2π
so let's say this is -2π
and then this one right over here would be -π
this of course is 0
then this is positive π
and then this right over here is 2π again
And then this could be 1
this could be 2
this could be -1
and this can be -2
Now let me copy this thing so I can use it for later
when I adjust this graph
So just this graph so let me copy
Alright so let's think about sin(x)
So what happens when sine is zero?
When sine is zero
Oh sorry, when x is zero, sin(0) is 0
Ok, I'll draw a little unit circle here for reference
This is what I like to do in my head
as I like to figure out the value
of the trigonometric functions
so this is x
this is y
Draw unit circle
And remember over here x is refering to the angle
So that's my unit circle radius 1
So then the angle is 0
sin is going to be the y-coordinate here
so sin(0) is a 0
when as sine increases
we go up all the way to sin(π/2) which is 1
so sin(π/2) is going to get you to 1
then sin(π) gets you to 0
sin(3π/2) gets you to -1
and then sin(2π) gets you back to 0
So if I were to graph this I'd look something like this
so this is between 0 and 2π it looks something like that
And we also want to go in the negative direction
and so as we go in the negative direction
as we go in the negative direction
so sin(-π/2) is -1
then you go back to -π and you go back to zero
-3π/2 you're going all the way back around like that
that gets you all the way back to sine is equal to 1
so sine is equal to 1
And then 2π you go back sine is back equalling 0
So the curve will look something like this
In the negative so as you go from
between 0 and -2π
And this is consistent with everything else we know about sine
the period of sin(x)
what you see here
you have a coefficient of 1 here
so the period is going to be 2π over the absolute value of 1
which is a little bit obvious it's just 2π
or you just see here that the period is 2π
It took 2π length to do our smallest repeating pattern
And what is the amplitude?
Well we vary between 1 and -1
The total difference between the minimum and maximum is 2
Half of that is 1
Or another way of thinking about it is
we vary 1 from our middle point
So that was pretty straight forward
Let's change it up a little bit
Now let's graph y=2sin(x)
So let me draw
let me put my little axis there
want to do it right under it
So what is going to happen
now that we have y = 2sin(x)
how is the graph going to change?
Well all we did is multiply this function by 2 so
whatever this value it takes on is going to be twice as high now
so 2 times 0 is 0
2 times 1 is now 2
2 times 1 is 2
2 times 0 is
let me be careful
2 times 1 is 2
that's a π/2
2 times 0 is a 0
2 times -1 is -2
2 times 0 is 0
so it looks something like this
between 0 and 2π
so it looks something like that
and we keep going in the negative direction
2 times -1 is -2
2 times 0 is 0
2 times 1 is 2
2 times 0 is 0
so in the negative direction it looks something like that
my best attempt to draw a relatively smooth curve
hopefully you get the idea
so it would look something like that
So what just happened?
Well the difference between the minimum
value and the maximum value just increased by a factor of 2
the total difference is 4, half of that difference is 2
So what is the amplitude here?
Well the amplitude is 2
you can view it as the absolute value of 2
well it's common sense
the amplitude here was 1 but now you're swaying from that middle position twice as far
because you're multiplying by 2
Now let's go back to
sin(x) and let's change it in a different way
Let's graph sin(-x)
so now let me once again put some graph paper here
And now my goal is to graph sin(-x)
y=sin(-x) so at least for the time being I've got rid of that 2 there
and I'm just going straight from sin(x) to sin(-x)
So let's think about how the values are going to work out
So when x is 0 this is still going to be sin(0) which is 0
But then what as x increases, what happens when x is π/2
the angle that we're inputting in to sine
we're going to have to multiply by this negative
so when x is π/2
we're really taking sin(-π/2)
but what's sin(-π/2) but we can see over here here it's -1
It's - 1
and then when x = π well sin(-π) we see this is 0
When x is 3π/2 well it's going to be sin(-3π/2)
which is 1
Once again when x is 2π it's going to be sin(-2π)
is 0
So notice what was happening as I was trying to graph between 0 and 2π
I kept referring to the points in the negative direction
so you can imagine taking this negative side
right over here between 0 and -2π
and then flipping it over to get this one right over here
that's what that -x seems to do
and by that same logic when we go in then negative direction
you say when x = -π/2 where you have the negative in front of it
so it's going to be sin(π/2) so it's going to be equal to 1
and you can flip this over the y-axis
so essentially what we have done
is we have flipped it we have
reflected the graph of sin(x) over the y-axis
So we have reflected it over the y-axis
This is the y-axis so hopefully you see that reflection
that's what that -x has done
So now let's think about kind of the combo
Having the 2 out the front and the -x right over there
so let me put the graph on the axis there one more time
And now let's try to do what was asked of us
So I'll do it in a new colour
I'll do it in blue
now let's graph y = 2sin(-x)
so based on everything we've done
how will this look
what are the transformations we will do?
If we are going from the original sin(x) to y=2sin(-x)
Well there're 2 ways of thinking about it
You could either take 2sin(x)
so here we multiply it by 2 to get double the amplitude
And you could say
well I'm going to flip it over to get the negative side of x
And so if you did that you'd get
so let me make it clear what I'm flipping
so if I took between 0 and -2π
and I flip it over
what used to be here you reflect it over the y-axis
and you now have
so we go negative first
then we go back to 0 and then it'll go positive
and then you get right over there
So all I did to get from 2sin(x) to 2sin(-x)
is I just reflected over the y-axis
and then of course what is between 0 and -2pi
you just have to look between 0 and 2pi
so it's going to go up
up and down
and make it even a little bit better
draw a little bit neater
and then down and then down and up
so it's a reflection of what was between 0 and 2π
so you see that right over here
or if you start with sin(-x) and you go to 2sin(-x)
notice what happens between sin(-x) and 2sin(-x)
What's the difference between this graph and this graph?
Well we just have twice the amplitude
we're multiplying this one by 2
and so you get twice the amplitude
And so the last thought or question I have for you
is how does the period of 2sin(-x)
how does that relate to the period of sin(x)
Well there's 2 ways of thinking about it
we could actually I'll you think about it for a second
Well there's 2 ways to thinking about it
you could refer to the graphs over here
or you can think about the formula which is hopefully a little intuitive right now
if you wanted to refer to the kind of classical formula
the period is going to be 2π
and you divide by the absolute value of the coefficient
to figure out how much faster you're going to get to 2π
So the absolute value of -1 is just 1
so you get 2π
So you get the exact same period as the period of sin(x)
And you see that
you complete one cycle every 2π ...
Now what is the difference?
Well the period's the same.
But remember this negative is not completely ignored.
It doesn't change the period but it does change how the graph looks
When you start getting increased x's
instead of sin being positive as it would be
in the case of a traditional sine function
here as x grows
you're taking the sine of -x
You're taking the sine of a negative angle
And that's why you start off having negative values of sine
and that's also another way
if you want to think about it
that it's just a reflection over the y-axis
of just sin(x)
These two are reflections and these two are reflections
this one is two times the amplitude of this one
and that one is two times the amplitude of that one