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Transforming sinusoidal graphs: vertical stretch & horizontal reflection


Video transcript

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