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good day students in this clip we're going to be reviewing some problems on graph graphs
of trig functions lets take a look at this here number one so it says DBA exam amplitude
and period of the graph so we have his graph below are makes you examine this closely what
kind of a trig function does this look like zero quick review remember your sign function
looks like an S and its positive the sign goes up like this that one. if its negative
goes in the other direction like this in your are your cosine function looks like a cup
so on. Of your cosine function in this positive positive cosine looks like this is registered
is positive sign in the red one is negative sign and for this one we see this, orientation
that is be positive cosine the positive cosine starts from the maximum okay and the negative
cosine starts from the minimum and goes in the opposite direction like that that is the
negative cosine okay let's take a look at this graph what does this look like this is
looks like the cups started from the maximum to this is a positive cosine function okay
Max for the amplitude amplitude is the displacement from the center to the Max or center to the
mean okay to the central position the midpoint from the tip to tip from the maximum points
to the minimum points this is a central position center and we have to maximum which is where
all the peaks Comstock that the maximum displacement from the central position and this is the
minimum displacement okay only have our answer Max any now notes that the amplitude is not
from the hundred displacement from the main to Max is the displacement from the center
to the Max are meant so i can either in it computes this distance from here to here or
I can computes this distance from here to hear okay so how many units displacement are
there from the center to the maximum or from the center the mean tha will tell us what
amplitude is an can clearly see that is for covered it up a little bit so this is for
units the case as our amplitude now to calculate hour. There are different ways you can calculator.
While complete cycles it. So I we had the maximum here from this maximum to the next
Max on this graph that will be our complete. Two can go from Max for maximum to maximum
lengths will tell you. Or you can go from minimum to minimum that we will tell you that.
Also okay let's do from Max to Max for maximum we can clearly see that the from this Max
to this Max these for units and I'm sorry units. We eight pi units so the period is
a product right cursor to write it down the amplitude with him before from the center
to the maximum is for units in the period from Max to Max is from your do it five is
units long to can clearly see that our answer is option letter the okay let's take a look at the number two we
asked to sketch the graph of these three functions and down basically say what your similarities
and differences are alright so lets start with that with the function number one we
have we're going to be graphing three cosine functions semester with the first one but
before we do that lets him right down the general form forty equation of trig function
so we have why is equal to a cosine bracket X minus H okay okay to that our what we going
to be using to generate a graphical dysfunction semester with the first one we have why and
we right this function in this format okay the coefficient of cosine is one quite equals
one cosine of the function of the X is one so we have bracket one times X minus H is
change the H2 A zero six minutes attracting anything from the text exponent zero + zero
can have this express so if nothing being added to it you have zero zero here okay now
let's determine what the new origin is well graph is going to be centered the origin is
going to be zero, zero you can see that our phase shifted zero and a vertical shift the
zero I lets figure out what our period is a period is to pi over b to buy ever be in
this problem be equals one so we have two pi equals our period In that tells is that
for checks is going to be equal to find okay in denominator calibration where using degrees
of one or I this in degrees we can have four takes equals pies one eighty-two two pieties
three hundred and sixty degrees I so i think at thing that's enough information to graph
this function oh yeah we also need and their amplitude our amplitude is a amplitude is
the absolute value of a celebrity computer the absolute value of one which equals right
now lets out go ahead and graph the function so we are starting from zero zero in our amplitude
is one so graphical one here.mil function can go in any political as low as one okay
so for chicks is equal to three hundred and sixty degrees so is ahead and a calibration
so we have off for chicks is equal to three sixty so we have great three sixteen two for
end up with but these for ticks here ninety-one eight seventy and three sixty so we have log
three sixty divided by four is ninety degrees isolated about the cosine functions we know
that a closer look like the cups let's graph that our graph the first one in red so we
have the starts from the maximum the center minimum center and right back to the maximum
cases that are cosine function looks like is connect the dots to purchase a sketch to
something like that there okay let's graph the second one we have armed function number
one graph has graph function number two limited in tightening cream so function to press right
quite close cosine four x in a general format hearsay can do why equals one cosine you noted
that these for now so four times X minus not it is subtracted from X is zero nothing is
being added to the entire functions those plus zero I can assign a new origin the origin
is going to be the opposite of the letter next to execute zero in in the numbered as
it added or attracting from thisterm which is your also power. The going to be all two
five... Three sixty this time three sixty into by the same thing to thirty-six silver
BBS Forza thirty-six silver four is going to be equal to ninety I and our officer that
tells is that for calibration for ticks is equal to ninety I have our amplitude amplitude
is the value a is probably can clearly see that a is one that the coefficient of cosine
so we have absolute value of one one so that's enough information to graph let's go ahead
and graph the function we still graphic cosine is to the first calibration for tick marks
equals ninety so we have to divide this ninety degrees of region into four so the quickest
way to do it with split down the center like this in the two house and go on to state the
house to create into another set of house okay so students can center like this that
anything like that okay right so we can graphic their so sign from the maximum Max center
main center Max as one complete. We can see that the much smaller than zero other one
they has a much compressed. So he graphic that cannot that pattern basically continues
for the entire region can write so that's our second function right there function number
two dollars graph function three in the see and make a comparison to for function three
we have why equals for cosine X now this right this function in this general format here
so we have why equals for that's not a cosine the coefficient of X is one minus zero + zero
because nothing is being added or subtracted to the X and nothing added to distance from
here now lets find our new origin the origin is zero zero opposite of the number next X
of the x-coordinate and and the other numbers y-coordinate hour. Is to find over the OT
used three sixty is the same test to buy radiance three sixty ribby which is one so for out
that's can give us the sixty degrees that tells is that for ticks is equal to the hundred
and sixty degrees okay so a stick will be ninety degrees and lastly we neither amplitude
our amplitude is the value of absolute value estates absolute value for is for units to
the going to have a displacement of four units up and down from the central position can
I we have all the information we need is go ahead and graph is maximum good to be at for
this time to the amplitude of four in our meeting mom affinity at the negative four
calibration will be the same as the first graph we do function number one so for tick
marks one to three four four tick marks being equal to three hundred and sixty degrees as
we can see here with the graph in our standard cosine function so stuff from the Max Max
center main center Max and we connect the dots can that there like that okay okay let's
go ahead and compare the graphs similarities and differences so this title of comparisons
what do we notice here. The green one is to the red one is one so we noted that the first
and the second one have the same amplitude okay function functions one and two have the
same amplitude amplitude of one right is obvious in the graph what about: functions one in
three the have the same. Right another this is the similarity similarity functions one
and three have the same.. Is that similarity that all three functions shared absolutely the all start
from their individual maximum ok than it will be the maxima so functions functions one to
and three start at your Max maximum now lets take a look at another example so for question number three there to graph
why equals can one half of X is equal to vertical symptoms in our sketch okay so first of all
to the where the asymptote integer at the two expressed can come using the quotient
identity okay so you can ask to find X over cosigned X so cosine basically tells us weightiest
symptoms of Tanis and we also have to recall that the town function at something like this
probably goes from negative infinity to infinity at every period. Right okay so analysis are
looking at can ask to graph one half X the graph cosine one half X now widely graph cosine one half X because that
is the function that tells us the locations of the asymptotes anytime the denominator
zero are cosine would have X have the value zero the town function has a vertical asymptote
okay so let's go ahead and graph graph back so we have why is equal to cosine one half
X that's right it in the form that makes it easy for us to determine its components so
have why is equal to one times cosine dropped one half times X minus zero was zero okay
now it is easy for us to find the relevant components that find the origin first the
new origin for cosine function is zero zero all hour. Is to buy over be and we can clearly
see that be is one half so we have to find over one half how do you divide to buy the
one half we're going to multiply the numerator by the reciprocal of the denominator which
is a fraction that is the same thing as dividing okay to have to find times to over one multiplied
across we have four answer. So this tells is that for ticks is equal to four pack seven
calibration process okay all the what CRR amplitude here although it's not useful for
this graph as you stated our amplitude is the value a which is how absolute value a
which is one right somebody sketch the cosine curve all that we do is just basically using
the cosine curve to determine where the asymptotes of our time of the related times curve is
located right to that's our maximum in public I'll take that out momentarily so that's maximin
celebrity and engrafted the calibration how the recalibrate our X axis we afford six one
two three four is going to be for pi oK going the other direction one two three four that
we negative four it's now it split our points for cosine function in every a cosine look
like the cups so you start from the maximum for positive cosine like that okay so start
from the maximum center next six center next is mean next six center next take Max right
going the other direction center minimum center Max now on which all my cosine function cosine
curve using these the dotted line to drive my certificate of the line you really have
to draw your cosine curve because it's just telling us what you think this around the
only point I care about are the points where cosine has a value of zero because the cosine
have the value zero you have the vertical symptoms for to can you see where the vertical
symptoms are the other roots of the cosine function rights to hear it hits zero let's
indicate where they are located we have ominously to hear here here and he okay so let's graph
the asymptotes first one second one second one third one and then we have for writing
that we follow the general pattern am that the town function follows as we illustrated
earlier scannable like he goes from negative infinity of to affinity for ever. So let's
us get without looks like so starting from the center. Go to infinity and then start
from negative infinity the center to centers to affinity in a from negative infinity back
to the ยท pattern continues forever to that's basically what the tan function looks like
from the left-hand side equals negative infinity to the center and right side the center goes
only to affinity than this one from the center towards okay so that's the graph this is the
graph of ten of one half X okay let's take a look at another example I it says select
equations that could represents the graph below kite center number four this is a trig
function that has phase shifted here to eleven to the right our consider a phase shift to
the right here that the closed is maximum so let's your you y-axis number over here
the question now is what kind of function is this what kind of trig function starts
from the maximum and starts to exhibit is the periodic behavior this is your cosine
function can't so we know that that's a cosine function one two three four five we know the
amplitude is five this positive because it starts from the maximum this where negative
cosine it will start from the minimum now the phase shifted the number of units that
it's shifted to the right okay said had a phase shift here so the question is how many
units with a shifted to the right of the better options we have to possible candidates either
to pi three four five over three so what is the accurate phase shift here will it has
to be smaller than pi over two the only degree measure thats smaller pi over two negative
zero pi over three so our phase shift is time over three okay so now we have enough information
to write down the quake the first equation of our graph is can be why equals the amplitude
is five five cosine part-time code 5 cosine the notice all the options have appeared want
to let the better about that is the facilitation X now we should private three units to the
right is that plus or minus five over three it's we do the opposite prices minus by over
three so can clearly see that on the first option is option C now on which other candidates
satisfy equation okay cycle the only class that, going to be over by the written identity
data you used to find another equivalent of function with this graph the co-function identity
WITH applicable to this problem is the identity that helps to express a cosine function and
have a sign function can so the identity element use is the co-function identity sign beta
the spy cosine data what does this mean this means that you have a cosine the have a sign
function and you shifted PI units to the left what you end up with is a cosine function
okay right of picoseconds right identity practicing negative of cosine negative cosine so the
one express cosine using negative cosine all you do is basically shift negative the cosine
function I'm units to the left right so initially F of our basic you have a C have cosine function
like this in the reflected to go like that anyway shifted PI units to the left you end
up with what you started with right sell their local function identity but this is one that's
applicable to this problem in the on the test I'll provide you with the be the the identity
okay so you have negative cosine is equal to the right so to this function tells me
if I would express cosine as a negative cosine at his have to Alpine to the humble okay to
that here in the negative cosine equivalent of on the result that we got sustainer look
like this why equals negative five cosine now that I have changed my cosine to negative
to compensate for that change to have exactly the same function that I started with I need
to add five to my can't ISA to finish it out for just that the this over one time it by
three top and bottom and then we have becomes a five cosine X minus the C- power to minus
the spread out minus PI over three plus refi over three and then you final answer is getting
negative five cosine X plus I over three) can clearly see our answer is option letter
the lets shift our attention to the next question so says which of the following graphs accurately
shows why equals negative to secant X so we have off for us into the trig functions and
the have from their asymptotes are determined either by Sino cosine carcass let's go over
those real quick this the fourth into the trig function tan cotangents secant and cosecant
Tangent is given by sign over cosine this tells is that cosine determines its asymptotes
could tangents is given by cosine #so signed basically determines where the asymptotes of cold tangent is secant
is given by one over cosigned the cosine determines where its asymptotes to its are just like
tan Okay and then cosecant is one over sine so sine determines where the asymptotes of
cool secant is our right now lets take a look at this option rehear the problem tells us
that's why X so the asymptotes of secant processing of secant is one over our cosine the cosine
function helps us to determining where using to are both have to be careful ureters and
negative okay so let's to be looking at the negative cosine so what does the negative
cosine function looks like remember the positive cosine is like a cup opening up to the negative
cosine is like a cup upside down right so the dotted line represents the function that
helps to determine the location of the symptoms to look at this right here is dotted line
is negative cosine this daughter line right here is positive cosine the is to negative
to cosine distance to cosine this dotted line here you know what is the second *** negative
to sign starting from the center and this one right here is sign so which of these actually
helps is determining why close to secant since secant negative secant independent negative
cosine this answer options day will be our current answer because it accurately determines
where our symptoms are which is basically the points when negative cosine executes the
X axes or have a value of zero right so there you have okay let's take a look at the next
question we asked to graph why equals can X + over two rights is the graph this distance
remember what the general appearance of the ten function looks like remember the chance
starting from the center just heads to infinity at the assented to the right. Negative incentive
negative infinity to this into to the left and exhibits the same behavior for ever okay
so the function that exhibits that increasing behavior from the left assented to the right
asymptotes option C or D or be okay in that circle that advocates outcome and eliminates
option a K this could be cotangents or negative so now we noted that is the phase shift here
express private to what does that mean we have why is equal to Times X Is Pirate to
What Kind of Phase Shift Is This Plus Poverty Ms. Use You Shifted Pirate to Units to the
Left Okay Pilot Two Units Shifted to the Left Is Left Direction Okay so Let's See Which
of Be Having Graft Poverty Is to the Left None of Them Have Only That Our Description
Then He Will Be Our Final Answer Okay to Poverty Units to the Left We Have One to Where One
Two Three so This Is a Private Three Is Rated and Are and over Here One Two Three Pilot
Three Is Right Center so Which of Them Is a Description of Uptown Curve That Shifted
Five over Two Units to the Left Okay Clearly See Is Option C Is the value Right Here Is
Pirate to Anything Shifted Time over Two Units to the Left Which Is Represented by Tan of
X Is Property Five Let's Take a Look at the Last Question Says the Graph of Why Equals
Cool Secant Text Have the Same Set of Asymptotes As Which of the Following Graphs to Remember
If We Express These Ones Using the Appropriate Identities We Can Know Where They Are Asymptotes
Are Located the Case so We Know That Why Equals Cool Secant X Can Be Expressed Using the Reciprocal
Identity As One over Sign X so If Any of These Function As a Sign in the Denominator We Use
the Quotient or Reciprocal Identities Then That Will Be Our Current Answer Right to Option
a White Equal Secant X Is Equal to the Written As One over Cosigned X Option DY = X Cannot
Be Expressed Using a Quotient Identity or the Reciprocal Identities so Labetalol Wise
Equal to Cool Secant to Ask Is Equal to One over Sign X Now Last but Not the Least Why
Is Equal to Cotangents X Is Equal to See What Is the Second Cosine over Sign X Okay so This
This Where This One We Would Have Assigned in the Denominator Seminary Can Be Here Be
but Which of Them Is the Current Answer Notice the Original Problem Calls for There to Be
Signed X in the Denominator with This #2 X Which Is a Problem but This #Exodus Is Exactly
What We Want so We Know That Option Season Answer to Final Result Is Option Letter D
This Function and Dysfunction Have Exactly the Same Asymptotes Protected a Look Alike
but Your Symptoms Are Identical Right so That's That the Thanks so Much for Taking the Time to
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