so today I'm going to teach you how to
graph the most common trigonometric
function sine and cosine without using a
calculator you actually only need to
know the transformation factors which
are the amplitude the period shape-shift
and vertical shift for a trig function to
be able to graph it and we can determine
all of these factors from the functions expression and
this video will also include some
examples where you can test your new knowledge so
let's get started
so this program is showing graph of this
sine function over here which is written in a
general form and the sliders here at the
top
let me change the variables in the function and
of course we also have a reset button that let
me reset all the sliders but now we're going
to try to make some connections between
the transformation factors and the
variables in this function let's start
by trying to change the a value a bit so
by increases the value of a we can see that the
amplitude of the function which is the
distance from one peak to the central line
seems to increase and that's also the
case because the amplitude for a trigonometric
function is equal to the absolute value
of this variable a the next variable is K
and we can see here that the period which
is a distance between two identical
points shorter with bigger values on k
and the connection between the period
and the K value is that the period is
equal to 2 pi divided by the absolute value of K
ok let's skip C for the time being and move on to D
The variable D is in charge
of the vertical shift of the function
so if you increase D you will see that
the function is raised a bit and if we
decrease it we can see that lowers the
whole function so now we only have the phase shift left
and only one variable left and the connection
between them is that if we increase the
variable C we will see that the function is
moved to the left and if we decrease it
we can see that the function moves to the right
so phase shift for a trigonometric
function is equal to minus C so with
that let's move on to some example i
will add the secret function, this red
function here i would like to pause the video
and try to determine the
transformation factors for this function
let's start by determining the amplitude
and the amplitude is the distance between
one of the peaks, the highest or the
lowest and the central line which in
this case is -2 and we can see here
that the highest peak is zero and the
difference between 0 and -2 is
2 and therefore the amplitude is going to be 2
let me plug that in right away
hm, yeah [Thinking really hard] that seems to match
pretty well, let's continue with another
transformation factor
we can observe in the graph that the
peaks are on the same distance from each
other, for both functions and that means
that the two functions have the same
period and we can also see that the
peak line up, so there's no phase
difference between the two functions either
but the red function seems to have been moved
two steps down, so therefore the D value
is going to be -2, so that's one
example determine a function
transformation factors, by just looking
at the graph, but I actually got two more
secret function, but at the same
time I dont want to bore you so
I will just link this program that's free to use
in the description, so you can play around with it
yourself and please let me know in the
comments tried it and managed to solve
the other secret functions
let's continue by doing some more
examples, our task is to try to determine
the Amplitude, period, the phase shift
and the vertical shift for these function
here in the table and to do that, we have to
use the rules that we just have discovered
and one important thing to remember is
that, we have showed that all these rules
apply to the sine function, but they do
in fact also apply to the cosine function
so that is a good thing to remember
so let's try to do the problems
our first function is to 2sin(3(x+1))
and if you use our rules we can
see that the amplitude is going to be 2
since that is the number in front of the
trigonometric function and the period is
going to be 2 pi divided by the absolute
value of K and K in this case is 3, so
the period is therefore going to be 2pi
divided by 3
the phase shift is the same as
negative of the C value and since our C value was 1
our phase shift is going to be -1
and we can also see that our D value is
going to be 0, since our function
don't have a second term to it and our
vertical shift is therefore also going
to be equal to 0, the next function is
a bit trickier, because we know how to
handle a trigonometric function that
are written in this form here, but this
function here is not exactly the same as
you can see, we are missing the K value
but we can fix that, let's try to rewrite
our function so that it matches our formula
lets se, sin(2x-2)
can be rewritten as sin(2(x-1))
now we have it written in the way want
and now we can quite easy again
with the help rules, determine the different
transformation factors let's see here
the amplitude is going to be 1, the period
is going to be 2 pi divided by 2, so pi
and the phase shift is going to be, lets see
C is -1, so the phase shift must be -(-1), that is 1
and the vertical shift is 1 here and for
the last one we need to remember that
it's not only the sine function these
rules apply to, it's also for the cosine
functions, so first of we need to rewrite it
as the one before and we get lets see now
2cos(3(x+1)) and now we can once again
just use the rules and determine the different
factors, so lets see now, we get that the
amplitude is going to be 2, the period is
going to be 2 pi divided by 3
the phase shift is going to be -1
as the c value is 1 and the vertical shift is
going to be 0, so that was everything I
had for this time
and I really hope you learnt something new
thanks for watching.
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