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 in the daily
low at both 10:00 a.m. and 10:00 p.m. and the highest temperatures are in the
afternoon right 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 then this right over here is my time in hours so that's
lowercase T time in hours and let's think about the range of temperatures
so the daily low temperatures around 3 degrees Celsius so let's actually
in the high is 18 so let's 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 it 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 midline of our trigonometric function is going
to be 10 point 5 degrees Celsius so let's let me draw the midline
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 the
3 degrees Celsius just like that so we're going to oscillate around this
midline we're going to hit the lows and 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 tea
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 where we're
assuming everything is in Johannesburg temperature T hours after I'm going to
say two hours after 10:00 a.m. and the reason why I'm picking 10 a.m.
is because we know that the temperature is right at the midline at 10:00 a.m. T
hours after 10:00 a.m. because if I want to graph f of T at T equals zero that
means we're at 10:00 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 we're back we're going to be back to 10 a.m. so our period is going to be 24
hours so let me put 24 hours there and then this is half ways 12 hours so what
happens after 12 hours after 12 hours we're back at 10 p.m. where we're back
at the 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 let's think
about what will happen as we go beyond as we start at 10 a.m. and go forward
so to go start at 10 a.m. and go for they tell us that the hottest part 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
six 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 look like
this and then our low so now we're at 10 p.m. and then you go six 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. after 10 a.m. you're going to be at your low temperature roughly right over
there and your curve your curve will look something something like this so
what would be before we even try to Model T of T let's what would be an
expression and obviously we keep going keep going like that we can even go you
know hours before 10:00 a.m. this is obviously just keeps on cycling on and
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 you say this is could be a sine or a cosine function actually
you could model it with either of them but it's always easiest to do
the simplest one which which function is essentially at its midline at its
midline when 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 midline of
just a sine function without it being shifted is is zero so sine of 0 is 0
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 this is going to be a little bit simpler
now let's think about the amplitude well how much do we vary what's our maximum
variance from our midline so here we are 7.5 above our midline here we are
7.5 below our midline 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 point 5 this is 7 point 5 so our amplitude looks like it's 7.5 and now
what is our period well we are we've already talked about it our period is
24 24 hours this distance right over here is 24 hours which makes complete
sense after 24 hours you got the same point in the day so we would divide 2 pi
by the period divided by 24 times T and if you forget hey you know divide 2 pi
by the period here you could just remind yourself that what are.what T value will
make us go from so when T is equal to 0 the whole argument to the sine function
is going to be 0 that's when we're over here and then when T is equal to 24 the
whole argument is going to be 2 pi so we would have made one rotation around the
unit circle if we think about the input into the sine function now we're almost
done if I were to just graph this this would be this would be have a midline
around 0 but we see that we've shifted 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
modelled it and we could simplify a little bit we could write this
as pi over 12 instead of 2 pi 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 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 let's just think about it let me just write it out so T of T
so T of T this is now we're modeling T hours after after midnight so we're
going to have the same amplitude we're just going to have the same variance
from the midline so it's going to be 7.5 times sine of which we do the same color
so you see what I'm changing and not changing times the sine of instead of
2 pi over 24 I'll just write pi over 12 instead of writing T I'm going to shift
T either to the to the right or the left and actually you could shift in either
direction because this is a periodic function we 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 ten point five plus ten point five 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 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 T 10 a.m. is 10 hours after
midnight so T capital T of 10 this is 10 hours after midnight should be equal to
should be equal to f of 0 because here the argument is hours after 10 a.m. so
this is 10 a.m. this right over here represents temperature temperature at
10:00 a.m. and this over here if because capital did this capital T function this
is hours after midnight this is also temperature at 10:00 a.m. so we want
T of 10 to be the same thing as f of 0 or a st. another way of thinking about
it when F of 0 this whole argument is 0 so we want this whole argument to be
0 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 0 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 point five so T of 10 should be F of zero so if we wanted to graph it we've
already answered their question we've if we put a 10 here the argument to
the sign becomes zero these two things are going to be equivalent but let's
actually graph this so T of 10 so if we're graphing capital T T of 10 so this
is 6 12 let's 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 0 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 and that makes sense because 0 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 if I this is going to be shifted by 10 so you can
get I'm just going to get it's going to be at 16 hours so let's see it's going
to look something something like that and of course it'll 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 why.