Hovedinnhold

## Constructing sinusoidal functions

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# Trig word problem: modeling daily temperature

## Video transkripsjon

Voiceover:In Johannesburg
in June, the daily low temperature is usually
around 3 degrees Celsius and the daily high temperature
is around 18 degrees Celsius. The temperature is typically
halfway between the daily high and the daily low
at both 10 a.m. and 10 p.m. and the highest temperatures
are in the afternoon. Write a trigonometric
function that models the temperature capital T in
Johannesburg lowercase t hours after midnight. So let's see if we can
start to think about what a graph might look
like of all of this. So this, let's say this
is our temperature axis in Celsius degrees. So that is temperature,
temperature, and I'm actually going to first, I'm going
to do two different functions. So that's my temperature axis. And this right over here
is my time in hours. So, that's lower case t, time in hours and lets think about the
range of temperatures. So the daily low temperature
is around 3 degrees Celsius. So lets actually, and
the high is 18, so lets make this right over
here 18, this right over here is 3 and we can also think about the midpoint between 18 and
3 that we hit at both 10 a.m. and 10 p.m. 18 plus 3 is 21 divided by 2 is 10.5. So the midpoint, or we can
say the mid line of our trigonometric function is going
to be 10.5 degrees Celsius. Let me draw the mid line. So we're going to
essentially oscillate around this right over here, we're
going to oscillate around this. The daily high is around
18 degrees Celsius. The daily high is around
18 degrees Celsius and the daily low is around 3 degrees, 3 degrees Celsius, just like that. So we're going to oscillate
around this mid line. We're going to hit the lows and the highs. Now to simplify things
because we hit this 10.5 degrees at 10 a.m. and
10 p.m., to simplify this I'm not going to tackle
their question that they want immediately, the hour in
terms of t hours after midnight. I'm going to define a new function. F of t, F of lowercase
t which is equal to the temperature, temperature
in Johannesburg, we're assuming everything is in Johannesburg. Temperature t hours
after, I'm going to say t hours after 10 a.m. The reason why I'm picking
10 a.m. is because we know that the temperature is
right at the mid line at 10 a.m. t hours after 10 a.m. Because if I want a graph
F of t at t equals zero that means we're at 10
a.m., that means that we're halfway between, they tell
us, we're halfway between the daily low and the daily high. Now what is the period of this trigonometric function going to be? Well after 24 hours
we're back to 10 am, so our period is going to be 24 hours. So we put 24 hours there and then this is, halfway is 12 hours. So what happens after 12 hours? After 12 hours we're back at 10 p.m. where we're back at midway
between our lows and our highs. And then after 24 hours
we're back at 10 a.m. again so those are going to be points on F of t. And now lets think about
what will happen as we go beyond, as we start at
10 a.m. and go forward. So as we start at 10 a.m.
and go forward they tell us that the hottest part,
the highest temperatures are in the afternoon. The afternoon is going to be around here. So we should be going up
in temperature and the highest point is actually
going to be halfway between these two so
it's going to be 6 hours after 10 a.m. which is 4 p.m. so that's going to be the high at 4 p.m. So let me draw a curve,
draw our curve like this. So it'll look like this. And then our low, so
now we're at 10 p.m. and then you go 6 hours
after 10 p.m. you're now at 4 a.m. which is going to be the low. This is 18 hours after
10 a.m. you're going to be at your low temperature,
roughly right over there. And your curve will look
something like this. So what would be, before we
even try to model T of t, what would be an
expression, and obviously we keep going like that we can
even go hours before 10 a.m. This is obviously, just keeps
on cycling on and on forever. Now, what would be an
expression for F of t and I encourage you once
again to pause the video and try to think about that. Well, one thing that you
say, well this could be a sine or a co-sine
function, actually you could model it with either
of them but it's always easiest to do the simplest one. Which function is
essentially at it's midline when the argument to the function is zero. Well the sine of zero is
zero and if we didn't shift this function up or down
the mid line of just a sine function, without
it being shifted, is zero. So sine of zero, zero and then sine begins to increase and oscillate like this. So it feels like sine is a good
candidate to model it with. Once again you could
model it with either but I have a feeling that this is
going to be a little bit simpler. Now lets think about the amplitude. Well how much do we
vary, what's our maximum variance from our mid line? So here we are 7.5 above our mid line. Here we are 7.5 below our mid line. So our amplitude is
7.5 and actually let me just do that in a different
color just so you see where things are coming from. So this is 7.5, this is
7.5, so our amplitude looks like it's 7.5. And now, what is our period? Well we've already talked
about it, our period is 24, 24 hours. This distance right over
here is 24 hours which makes completes sense,
after 24 hours you're at the same point in the day. So we would divide 2 pie by the period, divided by 24 times t. And if you forget, hey
you know divide 2 pie by the period here, you could
just remind yourself that what t value will make
us go from, so when t is equal to zero the
whole argument to the sine function is going to be zero,
that's when we're over here. And then when t is equal
to 24 the whole argument is going to be 2 pie. So we would have made one
rotation around the unit circle if you think about the
input into the sine function. Now we're almost done. If I were to just graph
this, this would be, have a mid line around
zero but we see that we've shifted everything up by 10.5. So we have to shift everything up by 10.5. Now, this is, we've just
successfully modeled it and we could simplify a
little bit we could write this as pie over 12
instead of 2 pie over 24. But this right over here
models the temperature in Johannesburg, t hours
after 10 a.m., after 10 a.m. That's not what they wanted,
they want us to model the temperature t hours after midnight. So what would T of t be? We're going to have to
shift this a little bit. So lets just think about it a little
bit, let me just write it out. So T of t, this is now,
we're modeling t hours after, after midnight. So we're going to have
this same amplitude. We're still going to have the
same variance from the mid line. So it's going to be 7.5
times sine of, I should be doing it in the same color so you see what I'm changing and not changing. Times the sine of,
instead of 2 pie over 24 I'll just write pie over
12, instead of writing t I'm going to shift t,
either to the right or left. Actually you can shift in
either direction because this is a periodic function
we're going to have to think about how much we're shifting it. So t is going to be plus or
minus something right over here. I'm going to shift it, plus 10.5. Plus 10.5, now this is
always a little bit, at least in my brain I have
to think about this in a lot of different ways
so that I can make sure that I'm shifting it
in the right direction. So here at 10 a.m. we were at this point. When t is equal to zero, this
is zero hours after 10 a.m. But in this function when is 10 a.m. Well in this function 10 a.m.,
let me write it this way. 10 a.m. is ten hours after midnight. So T, capital T of 10, this
is 10 hours after midnight should be equal to F of
zero because here the argument is hours after
10 a.m. so this is 10 a.m. this right over here
represents temperature, temperature at 10 a.m.
and this over here, cause capital T function, this
is hours after midnight. This is also temperature at 10 a.m. So we want T of 10 to be
the same thing as F of zero. Or another way of thinking
about it, when F of zero, this whole argument is zero. So we want this whole
argument to be zero when T is equal to 10. So how would we do that? Well this is t minus 10,
notice T of 10, you put a 10 here this whole thing becomes a zero. This whole thing becomes
zero and you're left with 10.5 and over here F
of zero well the same thing, this whole thing becomes
a zero and all you're left with over here is 10.5. So T of 10 should be F of zero. So if we wanted to graph it, we've already answered their question. If we put a 10 here,
the argument to the sine becomes zero, these two things
are going to be equivalent. But lets actually graph this. So T of 10, so if we're
graphing capital T, T of 10 so this is 6, 12 lets
see so this is maybe, so 10 is going to be
someplace around here. So T of 10 is going to be
the same thing as F of zero. So it's going to be like
that and then it's just going to, and then we've
essentially just shifted everything to the right,
everything to the right by 10. That makes sense because
zero after whatever hours you are after 10 a.m. is
going to be 10 more hours to get to that same point after midnight. So your curve is going
to look, so this is going to be shifted by 10, this
is going to be shifted by 10 and your curve is
going to look something like, let me see my, this
is going to be shifted by 10 so you're going to get, this
is going to be at 16 hours, so it's going to look something like that. And of course it will keep
oscillating like that. So essentially we just have
to shift it to the right by 10, the argument we
have to replace t with t minus 10 to do it and
this was the logic y.