How To Graph Trigonometric Functions - Free Educational videos for Students in K-12

How To Graph Trigonometric Functions - By The Organic Chemistry Tutor

Transcript


00:00	Now let's talk about graphing trig and metric functions .
00:04	Let's start with to sign function . Synnex syntax is
00:13	basically a sign of sort of function . It's a
00:15	sine wave and that's how it looks like at least
00:19	that's one period . This ends at two Pi that's
00:22	one cycle of the wave . Now let's say if
00:25	you put a negative in front of the sine function
00:30	it's going to flip over the X . Axis .
00:33	So instead of going up initially it's going to start
00:35	from the origin , it's going to go down and
00:37	then back up and then back down . So that's
00:42	the shape of sign and negative sign . Now keep
00:44	in mind this way it keeps on going forever in
00:47	both directions . But for the course of this lesson
00:52	I'm going to focus on graph in one period which
00:55	is basically one cycle of the wave . Now .
01:04	What about the graphs of cosign X . And negative
01:08	cosine X . Co sign starts at the top whereas
01:14	sign starts at the center so that's one period of
01:19	the coastline . We've , let me do that a
01:21	little bit better but it can continue going on forever
01:28	. Yeah , negative CO sign starts at the bottom
01:32	. It goes up to the middle and then goes
01:35	up and then back down . So that's the graph
01:38	of one period of negative co sign . So this
01:42	is one cycle . Now let's go back to the
01:46	sine graph . Let's draw two cycles of this graph
01:54	. So one cycle you need to Break it up
01:59	into four useful points . one cycle is two pi
02:06	. You want to break that up into four points
02:08	such as part of a two pi and three private
02:11	too . Now if we want another period let's add
02:15	two pi to it . So we want to go
02:17	to four pi In between two pine for Pious Street
02:20	pipe In between two pi and three pi . It's
02:24	five pi over two . You add these two ,
02:27	then divide by two . If we had three pi
02:29	and four priced at seven pie And then divided by
02:32	two . We get 7/2 . Now sign starts at
02:36	the center and then it's going to go up back
02:40	to the middle , down , back to the middle
02:43	. So that's one cycle of the sine wave and
02:48	then it's going to go back up back to the
02:50	middle down and then back to the middle . That's
02:54	why it's helpful to plot the points first before putting
02:58	everything else . If you break up each cycle into
03:02	five key points which equates to four intervals , it's
03:08	going to be easier to graph the sine wave .
03:10	Let's do the same for co sign . Let's graph
03:20	two periods of the coastline wave . So one period
03:24	is going to be two pi two periods for parts
03:28	for each period or each cycle . Break it up
03:30	into five points which is four intervals . The first
03:37	point , by the way is the origin it zero
03:42	. So these points will be the same as the
03:45	graph is signed . Mhm . Now we know that
03:53	co sign starts at the top . It's going to
03:56	go back to the middle and then to the bottom
03:59	, back to the middle and then to the top
04:02	and it's going to alternate . So it's going to
04:09	look something like this . So that's one cycle and
04:20	here's the second cycle . So that's two cycles of
04:24	the coastline wave . Now let's talk about the amplitude
04:28	of the sine wave . The generic formula is a
04:34	sign bx plus C plus t . Now we're going
04:40	to focus on a a the number in front of
04:44	sign is the amplitude . So in this case the
04:47	amplitude Is equal to one . So when you graph
04:52	the sine wave and you plot your four points of
04:56	interest for one full cycle , The amplitude is going
05:01	to be one . So it's going to vary from
05:04	1 2 -1 . So we know sign starts at
05:08	the center , it's going to go to the top
05:11	, backs in middle and then to the bottom and
05:13	then back to the middle . So it's going to
05:16	look like this And we know the period is two
05:19	pi . Now what if we wanted to graph to
05:24	Synnex ? So if we increase the amplitude , this
05:29	graph is going to stretch vertically , So it's going
05:41	to vary from 2 to -2 . By the way
05:44	, this is the amplitude , it's a distance between
05:50	the midline of the sine wave and the highest point
05:59	. Now let's plot in one period . So this
06:02	is going to be two pi . So once again
06:05	sign is going to start at the middle . Then
06:08	it's going to go up back to the middle and
06:11	then down and then back to the middle . So
06:18	it's going to look like that . And so the
06:21	amplitude tells you how much it's gonna stretch or compress
06:24	vertically . Consider the equation Why is equal to -3
06:30	coast in X . What is the amplitude of this
06:35	function ? The amplitude is always a positive number so
06:40	you ignore the negative sign and it's going to be
06:42	three . The amplitude is the absolute value of a
06:45	the number in front of coastline . Now let's go
06:48	ahead and graphic Let's plot one period . So let's
06:56	break it up into five points or four intervals now
07:02	the amplitude history . So we need to vary the
07:07	sine graph or rather the coastline graph from negative 323
07:12	So co sign typically starts at the top but we
07:15	have negative co sign . So it's going to start
07:17	from the bottom . Then it's going to go to
07:20	the middle to the top , back to the middle
07:23	and then back to the bottom . So that's how
07:31	we can graph one cycle of negative three . Co
07:35	. Sign X . Now keep in mind this graph
07:39	can keep on going forever in both directions . So
07:43	let's say if you want to write the domain and
07:45	range of this coastline graph , the domain for a
07:49	sign and coastline graphs will always be the same .
07:52	It's our phone numbers . The range is based on
07:56	the amplitude , the lowest Y value is negative three
07:59	, the highest Y value history . So that's how
08:03	you can right the domain and range of this particular
08:06	coastline graph . Now let's talk about finding the period
08:14	. So given this sine function , a sign bx
08:18	. We know A represents the amplitude . Now B
08:21	is not the period itself , but it's used to
08:24	find the period . The period is two Pi divided
08:29	by B . So in the case of Sine XB
08:33	was equal to one . So the period was two
08:36	pi divided by one . Now let's go ahead and
08:40	graph these two functions sign X And sign two x
08:48	. Let's see what effect he has on a graph
08:53	. Now we know the general shape of syntax .
08:55	It has a period of two pi and for the
09:00	most part it looks like this . Now if B
09:05	is equal to two in this example the period is
09:08	going to be two pi divided by B . So
09:10	the period is pie . So therefore it's going to
09:15	do one full cycle in less time , so to
09:21	speak . So what happens is the graph , it
09:26	shrinks horizontally . So one full cycle occurs in one
09:33	part . two cycles occur in two pi . Here's
09:39	another example . Go ahead and graph dysfunction to sign
09:43	one half X . So first we need to find
09:47	the amplitude . The amplitude is the number in front
09:50	of sign . That's too . The period is two
09:53	pi over B . Where B . Is the number
09:55	in front of X . So in this case is
09:57	one half , two Pi divided by 1/2 is four
10:00	pi . Mhm . So this one is going to
10:05	stretch horizontally . The amplitude is too And the period
10:12	is four pi but we need to break it up
10:14	into four intervals . Yeah , so that's one pi
10:18	to pie three pi and four pi sign starts at
10:23	the center . Then it goes up back to the
10:26	middle , down and then back to the middle .
10:32	So we're gonna have a graph that looks like that
10:35	. So if you have a fraction what's going to
10:36	happen is it's going to stretch horizontally . Let's try
10:41	another example . Let's graph four co sign pi X
10:51	. So first identify the amplitude and the period The
10:55	amplitude is simply for in this example And the period
11:00	is two pi over B . In this case would
11:03	be is the number in front of X . So
11:06	be a spot . two Pi divided by Pi is
11:09	too . So that's the period in this example .
11:13	So let's go ahead and make a graph . So
11:21	the amplitude is for So it's going to vary from
11:28	four and negative for the period is to so too
11:34	should be about here and we need to break it
11:40	into four parts . So this is one , one
11:43	half and then between one and two you add them
11:46	up one plus two or three , then The average
11:49	it or you divided by two , so it's 3/2
11:52	. So those are the four points of interests co
11:56	sign starts at the top , then it's going to
11:58	go to the middle and then back to the bottom
12:02	to the middle and to the top . So we're
12:05	gonna have a graph that looks like this , that's
12:09	one cycle . And if we wish to extend it
12:12	to draw another cycle This is going to be three
12:16	. Next one is 2.5 or 5/2 and then three
12:21	plus 47 but then divided by two . So 3.5
12:24	is 7/2 . The next point is going to be
12:29	at the middle and then back to the bottom ,
12:31	back to the middle and then to the top .
12:38	And that's it . So that's how you can grab
12:40	four co signed pie attacks . So when you find
12:43	your period , make sure you put that first on
12:46	the X axis and then break it into four intervals
12:49	. Now , what is the domain and range of
12:52	this function As you recall ? The domain for any
12:56	sign or co sign wave is our phone numbers .
12:59	The range is from negative 4-4 . It's from the
13:04	lowest Y value to the highest Y value . Now
13:09	let's talk about what to do when there's a vertical
13:11	shift let's say if you wish to graph sign X
13:16	. Plus string . So the vertical shift history The
13:21	amplitude is one . So what you want to do
13:25	first is you want to plot the vertical shift .
13:32	So at three I'm going to draw a horizontal line
13:36	. That's gonna be the new center of the graph
13:39	. The amplitude is one . So sign . It's
13:43	going to vary one unit higher than the midline and
13:46	wanting it lower than it . So it's gonna vary
13:48	between two and 4 . Now we're still going to
13:51	plot just one period . So let's write our four
13:54	key points sign , starts at the top and then
14:00	it goes to the middle . Actually I take that
14:03	back , Science starts at the middle and then it
14:05	goes to the top and then back to the middle
14:08	to the bottom and then back to the middle .
14:10	So this would be one sine wave . So that's
14:13	how you can grab Synnex plus during let's try another
14:17	example , Let's Graph two periods of to co sign
14:26	X minus one . So this is going to be
14:38	one cycle and two cycles . But let's start with
14:42	the first cycle . So the midline is that negative
14:47	1 ? Now the amplitude is too So we got
14:55	to go up two units And down two units .
15:01	Now co sign we'll start at the top and then
15:06	it's gonna go to the middle , back to the
15:08	bottom and vice versa . Now we need to plot
15:12	one more cycle . So this is pie and this
15:17	is street pipe , so it's going to go back
15:20	to the middle and then to the bottom back to
15:23	the middle and to the top . So that's how
15:26	we can graph to co sign periods . Now what
15:34	is the range for this graph ? Notice the lowest
15:37	, why values that -3 , but the highest is
15:40	that one . So the range , It's from -321
15:48	. Let's go ahead and grab this one , negative
15:50	three . Sign X plus four so feel free to
15:55	pause the video . Actually , let's also let's change
15:58	it a bit , Let's make it 1 3rd X
16:02	plus four . The majority of the graph will be
16:07	above the X axis . So let's draw the center
16:17	line at 4 . 1 the amplitude history , so
16:23	we're gonna have to go up 34 plus three is
16:25	seven And then down three starting from 4 , 4
16:28	ministries one . So the range Is going to be
16:33	from 1 - seven . Now let's find the period
16:37	. We know the period is two pi divided by
16:40	B And be as 1/3 . So it's two pi
16:43	divided by one third , So it's equal to six
16:46	parts . And let's break into four points , half
16:50	of six , prior ST pie , Half of three
16:52	pipe , It's three Part 2 . And if you
16:55	multiply this number by three , it will give us
16:58	to this point which is 95 - two . Now
17:03	we know that sign starts epicenter positive sign will go
17:08	up initially the negative sign , we'll go down and
17:11	then it's going to go back to the middle And
17:14	then to the top at seven and then back so
17:17	little so that's how you can plot negative three .
17:22	Sign one third . X plus four . Now let's
17:27	talk about how to graph this function . Sign X
17:32	minus pi divided by two . How can we do
17:35	? So ? So considering the generic formula A .
17:40	Sign Bx plus C plus T . Any time there's
17:48	a C value , there's a face shift which means
17:52	that the graph is going to shift either to the
17:54	right or to the left . And so you want
17:57	to find the face shit because sign won't start at
17:59	the origin in this case . So to find the
18:02	phase shift set the inside equal to zero and soft
18:06	racks . So when you set Bx plus C equals
18:08	zero . X . Is going to equal negative C
18:11	divided by B . And this is your face shit
18:14	. That's where it starts on the X . Axis
18:23	. So let's set X -9/2 equal to zero .
18:27	So we can see X . Is that private too
18:30	? So that's where the sine wave is going to
18:31	start . Now let's go ahead and graphic the amplitude
18:37	is one and the period is two pi over one
18:40	. So it's two pi but first plot pi over
18:44	two because that's where the phase shift is . And
18:49	then what you want to do is add one period
18:52	to face shift . So you're adding to pilot to
18:54	private too . two Pi is the same as for
18:57	private tune . So this will give you five pi
19:00	over two . So this is going to be three
19:07	part of it too and you want to break it
19:09	into five key points . This is one private to
19:12	in between one and 3 is too . To privacy
19:15	was part In between three part of it too and
19:18	five party too . We have four private too ,
19:20	Which reduces to two pi . Mhm . Now the
19:29	amplitude is one , so it's going to vary from
19:31	one . A negative one . Now sign starts at
19:35	the middle but we're not going to start the origin
19:38	in this example we're going to start at the phase
19:41	shift which is private . E positive sign . It's
19:44	going to go up , negative sign is gonna go
19:46	down first , it's a negative sign will be like
19:48	this positive sign will have that shape and then at
19:53	two pie it's gonna have a Y value of negative
19:55	one . And that five private to it's going to
19:57	be back on the X . Axis . So that's
20:01	how you can plot this particular sine wave with a
20:03	face shift . Now let's try another example let's say
20:09	if you want to plot tune sign X minus prior
20:15	before plus three . So we have a vertical shift
20:24	of three . An amplitude of two . The number
20:28	of front of exes one , so two pi ,
20:30	everyone is two pi The period is still two pi
20:32	but we do have a facetious . So if we
20:35	set the inside equal to zero , The phase shift
20:37	is positive pi over four . The majority graph will
20:41	be above the X axis . So we're going to
20:44	plot it up there . So let's plot the Midline
20:48	1st or the center line Which is at three .
20:53	The amplitude is too , so we need to travel
20:56	through two units above the center line which will take
20:58	us to 53 plus two is five and then two
21:01	units down three minus two is one . So the
21:04	graph is going to vary from 1-5 and that's the
21:07	range of the sine function . Now the phase shift
21:12	is going to start at pi over four . That's
21:15	where the sine wave is going to start . And
21:18	if we add one period to that , the period
21:21	is two pi to private . One is the same
21:24	as a pie . Before we need to get common
21:26	denominators . So if we are these two numbers ,
21:30	this will give us nine pi over four . So
21:36	that's where the first period will end , The midpoint
21:42	between one and 9 is five and the midpoint between
21:45	one and 5 is three And between five and 9
21:50	is seven . Now we can graph it . So
21:58	let's start with the face . If sign is going
22:01	to start in the middle and then it's going to
22:03	go up back to the middle and then down and
22:07	then back to the middle . So this is the
22:11	graph of just one period .

Summarizer

DESCRIPTION:

This trigonometry video tutorial explains how to graph sine and cosine functions using transformations, horizontal shifts / phase shifts, vertical shifts, amplitude, and the period of the sinusoidal function. This video contains many examples and practice problems on graphing trigonometric functions for you to master this topic.

OVERVIEW:

How To Graph Trigonometric Functions is a free educational video by The Organic Chemistry Tutor.

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