Graphs of trig functions
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Example: Graph, domain, and range of sine function
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Example: Graph of cosine
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Example: Intersection of sine and cosine
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Example: Amplitude and period
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Example: Amplitude and period transformations
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Example: Amplitude and period cosine transformations
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Example: Figure out the trig function
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Graphs of sine and cosine
Example: Amplitude and period transformations Understanding how the amplitude and period changes as coefficients change.
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- 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
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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