Stony Brook MAT 123 Fall 2015
Lecture 25: A garden of graphs
November 18, 2015

Start   We were talking about graphing last time so we will do more on graphing. As I said, Professor Sutherland and I realized that a lot people are uncomfortable with graphing. Which is understandable, because schools use graphing calculators for graphing, and a graph is just some picture that comes out from your calculator.
I kind of want you to think about graphing a little more. So remember I talked about the absolute value of sine, the absolute value of cosine, their graphs, we can make a graph library. We talked about what is sin(1/x). sin(x) remember looks like this.
0:47Then, we have sine of 1 over x. that gets a little tricky because 1/x behaves badly as x gets close to zero. First of all,
1:07what happens when x is very big? When x is very big, 1 over x is going to be close to zero.
So when x is a million, 1 over x is 1 over a million. Close to zero.
So way out here, we really almost taking the sine of zero and the sin(0)=0. So, far out is just
1:30how the value of zero and this is kind switch the sine. It sort approaches the axis like this.
The end behavior will approach zero. Now what happens as you get close to x? close to the x axis. So now when x is small, when x is one this is the sine one 1, 1 radian.
When x is 1/2 the sine of 2, when x is 1/3 the sine of 3, when x is 1 over 1000
2:04the sine of one thousand. So to get the sine of a bigger and bigger number, what happens to the sine?
It just goes up and down. The sine of 1000 is not necesarely bigger than sine of 999. It depends on the radians.
Since we keep switching up and down as we get different numbers and as x gets closer and closer to zero and 1/x is getting bigger and bigger so this is squeezing on the side.
2:32It starts doing this. What happens is it wiggles like crazy as it goes into zero. It is the same on the other side and then it goes like that. Pretty looking graph. Maybe it it was the same in both directions.
The sin(1/x) is very strange looking and that is because 1 over x behaves kind in a funny way. So, why does it do this? When x is very large, 1 over x approaches zero so
3:04you are looking at basically sine of zero ok? So when x is 10 you are looking at sin(1/10), when x is 1000 you are looking at sin(1/1000), so for x big it is close to sine of zero, but it is going to go up and down so sometimes you get negative values and sometimes you get positive values, approaches zero. When x gets very close to 0, 1 over x gets very large. So, the sine will start alternating back and forth as you
3:34come up with other numbers. Sine of, let me do it in the calculator sin(100), sin(101), equals... I dunno. So you cannot use a calculator but I can use a calculator so you know, I am in front of the room. Kind of weird I have half on my audience today.
So, sin(100)= -.5 and sin(101)= +.45, sin(102), sin(103), sin(104) is negative again.
4:18So every a few numbers it switches from positive to negative. So when x gets closer to zero, remember this number gets bigger and bigger. You take the sine of bigger
4:30and bigger numbers which sometimes is positive and sometimes is negative. That is why is going up and down. Remember you are getting close to zero these values keep switching the signs. That is why is like that. So that is a fun graph. We love to use fun graphs, we can ask you questions sin(1/x). Why? Because other stuff is easy and I have to ask you the hardest ones too. Ok everyone got the idea? I am not asking you
5:06to graph sin(1/x), I am asking you to understand what sin(1/x) looks like. If we ask you to graph that on the exam that would be very mean.
Ok, that was sine(1/x). What about 1/sin(x)?
Also known as cosecant x. Let's figure it out!
5:41So that is our sine graph again. So what is 1 over that look like?
The first thing is remember that the sine of pi is zero.
Sin(2pi)=0. What happens when you do 1 over zero? Cannot get over zero right?
6:11It is undefined. So we will have a vertical asymptote at all places when sine is zero.
And that one in the middle, OK?
6:31How do I write the middle? At pi/2, the sine of pi/2 is 1. So here is pi/2. 1 over 1 is also one so we get that value. So now what happens between zero an pi/2, you get 1 over these numbers.
These are numbers between zero and 1. So when you flip them, they get bigger than one.
7:00In other words say you have pi/6. So 1 over half is 2. You guys see that pi/6 sine is 1/2 so 1 over 1/2 is 2 and one over radical 3 which is about 0.6 you do one over that. You get a number bigger than 1 and it is going to approach the vertical asymptote. The other side, you do the same thing. So it kind looks like a parabola but it is not. On this side, bigger values going
7:48down to bigger once. So this graphs is the same as this one. So if the sine graphs looks
8:02like that then you broke it and it is this way instead. So you take this "u" and you turn it up side down and then you take this and you turn it up side down and you get this. So that is what the cosecant 1/sin(x) looks like. So 1 over graph is another entertaining type of graph.
because the sine has places where it is is zero.
Alright. Let's see if you guys can figure out what the 1 over cosine graph looks like.
8:42We got y=cos x. Remember it kind look like that. pi/2, 3pi/2, 2pi, pi. ok!
9:03That is cosine x. Why don't you take a minute and see if you can figure out what secant graph looks like.
So cosine and sine are the same graph. They are just shifted by pi/2 (90 degrees). So 1 over sine and 1 over cosine also be shifted. So first you want to find where it is zero. So it is zero at -pi/2, -3pi/3, and +pi/2 and +3pi/2.
9:38So on. So you will have vertical asymptote there. ok!
Is equal to one at zero so 1/1 is going to be one. It is going to be -1 at pi. So 1 over
10:02-1 is -1. And you get "u" shape again. This one approaching infinity and down to negative infinity. Again these going toward infinity is these as they get close to zero and if you do 1 over a small number you get a big number. This is something that you have to really grasp at the beginning of calculus. 1 over a very small number is very big number, 1 over a very big number is a very small number. Ok? So as x gets very large, approaches infinity
10:361 over x approaches to zero. When x gets very close to zero, 1 over x goes to infinity You got this? You can draw it nice and pretty, maybe some color, stars on it.
This is y=cos x, this is y= 1/cos(x) also known as sec(x). So that is the cosecant and
11:11the secant graphs. We don't do much with them, you should just understand where they come from and what they look like.
So far so good? I'm gonna cover this up in a second.
11:31Let's go back to this 1/x concept just for another minute.
This whole ties in something called the limit that you will see at the beginning of calculus.
The graph of y= 1/x looks like this. So notice what happens to 1/x. This is (1,1), That is a hyperbola by the way. As x gets very large this graph appraoches the
12:06x-axis as it gets close to zero.
Let's think about what happens when x gets large.
As x gets larger and larger this is 1/1, 1/10, 1/100, 1/1000 and so on. This number is getting very small. 1/1000 is not a very big number.
12:35Its 0.001. So when x is a trillion, this is one trillionTH.
So when x is very large the y value is very close to zero.
It is 1 over that number. And, when x gets very small, so what do me mean by small?
when x is 1/10 y is 10, when x is 1/100 y is 100, when x is 1/1000 y is 1000 and so on.
13:06So now when x gets very small, y is very large. That is why blows up here.
So you want to get that principle down.
Imagine the sin(x) graph we had. The sin(x) graph kind of mirrors this graph. Imagine the second. make sense?
The sin x graph looks kind of mirroring this graph.
13:35Imagine the second. does that make sense?
This is a crucial thing to get in your head when you have to do these kinds of graphs.
You will see a bunch of these in a calculus class, 1/x, 1/x+1, 1/x-1, things like that.
You want to get an idea what the basic shape of that graph looks like.
14:03How about y=x+sinx?
Remember what sin x does?
Sine x just goes up to 1 and down to -1 and back and forth.
Doesn't really do much anything.
14:31x in the mean time gets bigger and bigger.
y=x So when x is 5, y is 5 right?
So, what happens is you take x and you add some values to it, it will go up and down from 1 to -1.
Sometimes it can be add to it, you never add more than 1 Sometimes you can subtract from it, subtract more than one.
So if you think about it,
15:02this is the graph y = x.
So what's going to happen is, sort of imagine y=x line.
You are going to add to it. So you going to come up above it and you going to subtract a bit, are going to add a bit.
15:30you subtract. You can sort of follow the line y=x.
Remember the dotted line has nothing to do with the graph. So the graph just keeps going higher and higher.
Sometimes it goes under y=x and sometimes goes above y=x.
Just depends on whether you are adding or subtracting to it.
If you are adding or subtracting sin x. Make sense? That is kind of a fun graph! Sure, you can put this in the calculator and see what the graph look like.
16:03So if I had -x+sin x, it will be down here It kind of snakes along the line y=x.
How are we doing so far?
This is actually adding two different graphs together.
16:49Let's see if you can figure out what sin(x) + cos(x) looks like?
17:09Alright this will definitely be on the final.
Yeah, I'm gonna put this on the final, no doubt. Sure.
Lets just take some values.
17:37Alright, what happens at zero?
Well sin(x) is zero, cos(x) is 1 and 1.
At pi/6, this is 1/2.
This is radical 3 over 2.
So when you add 1/2 and radical 3 over 2 you get about 1.4, doesn't really matter.
18:07Well, let's make that not 1.4 let's make that 1.3.
Because if you add these two you you get about 1.4.
Then if you add, at pi/3 this is radical 3 over 2, this is 1/2.
If you are adding the same thing you add before (1.3)
18:31and at pi/2 this is one and this is zero.
So when you do sine and cosine shape, we just go up a bit and down a bit You can certainly figure out the first part of the graph It would be here at 1, it will go up and go down at pi/2.
Now what happens?
Now cosine is going to be negative and sine still going to be
19:00positive when you start subtracting values.
So If we had,
19:33sine of 2pi/3 is radical 3 over 2 Cosine of 2pi/3 is negative 1/2 so this is about .3 3pi/4 this is radical 3 over 2 and negative radical 3 over 2, zero.
20:04Let me graph, it is zero somewhere around there.
Think about now this is going to get smaller when you subtract more when we get down to here this is 0, 1, -1.
So this sort of kind of behavior. I am going to leave up the graph now.
20:32This is going to behave a lot like what a sine and cosine graphs looks like.
We don't really need points It is still looking at something like that.
21:05So whats another way we could have done this without plotting the points.
I am going to draw them on the same set of axis.
So that is sine. First graph. This is sin x
21:30and this is cosine x.
So if we want to add them together, well here we adding zero and here we are adding 1 So we are going to be at 1 And now we are adding something that is coming down to 1 and something that is going toward 1.
So if we are adding a little bit to the graph and right here sine will come down to radical 2 over 2 and cosine will go up to radical 2 over 2 And that would be getting the biggest value we can get.
22:05And from over there sine is getting bigger and cosine is getting smaller, cosine is heading down to zero.
Now sine is getting smaller and cosine is now a negative value Right at this point.
So we get to out most negative value.
So, it is going to come down to there.
22:38So this is opposite radical 2, negative radical 2.
This curve will go up again.
So kind of like sine sort-of-graph except it has a different top and a different bottom.
So how do I know the maximum is? the maximum is where i am adding together biggest possible combination of sine and cosine.
where it adds up to radical two.
23:03And the negative value, negative radical 2.
How about non-trig graphs for a while. Those would be good right?
23:45These questions, this is a question a lot of people have trouble with on the minimum competence.
Lets ask if a graph looks like this.
24:10I didn't graph anything in particular it is just a graph.
For how many values, how many values of x where y=1?
How do you figure that out?
24:32I am just waiting. (Audience: 5) How do we know that there are 5 values?
well, you go to y=1 and you just draw an horizontal line.
So where y equals 1 intersects the graph.
Look what the graph is telling you.
The graph is giving you information of the value of y.
You got 1, 2, 3, 4, 5 intersections.
Lets say where y=2, you have 4 intersections.
25:02This is where it is equal to 1/2, 3 intersections.
So we gave a problem like this on web assign and we gave problems like this on the minimum competence.
A lot of people have trouble with it.
So if you want to know for what value, for how many values of x
25:30does f(x)=1?
What you do is, you draw a horizontal line y=1 and see how many times intersect the graph.
Make sense?
What if I said for how many values does x =1?
How many values of f(x) does x =1, the reverse of that.
Then you draw a vertical line and in fact on functions you can only get either one or zero.
26:03Ok? Because you cannot more that one value otherwise it is not a function.
So if they say for how many values of f(x) x=2, x=3.
Always is going to come out 1 or is going to come out zero if the graph has a hole. a gap.
Lets just practice a little more with logs and exponential kind of stuff.
So remember what y=e^x looked like?
26:34y=e^x looks like something like that. Ok?
It goes through point (0,1) Why does it goes through (0,1)?
Well, e^0 is 1. In fact any number, other than zero, raised to zero will give you 1.
So if I had to graph of 2^x it would be a little shallow
27:01here and a little flatter there ok?
So this is e^x and this would be 2^x and 3^x goes the other way, 3^x would be inside on this side.
Why is that true?
well just pick a number.
When x=1 this is 2, this is 3. And "e" is at 2.7. ok?
But all these graphs have the same shape.
27:32When x is equal to a large number they get very big and very fast.
Because 2^10 for example is 1024 so when x is 10 it is way up there, 1024.
So it blows up very quickly.
And when it is on this side, we have 1/1024 so gets very close to zero.
Notice, when x is a positive number we get a big positive number, when
28:02x is a negative number you get a very small positive number.
Cannot be zero.
What value, lets see 3^x.
What would I raise x to, to get zero?
You cannot raise to anything.
If I do 3^0 is 1, 3 to anything grater than zero is a big number.
3 to anything less that zero is a small number but positive
28:30If I do a fraction, if I do a negative, it does not matter I cannot get zero This graph will always going to have the x-axis be the horizontal asymptote.
So of course I can now give an asymptote [unintelligible] For example y=e^x +1.
It'd look like that, OK?
29:03the whole graph will shift up one.
And if I have e^x-1 the graph will shift down 1.
So we have y=e^x+k, k is going to be the our horizontal asymptote you sure?
How about
29:31now if you know what e^x looks like, let's remember again what natural log of x looks like So e^x looks like this, natural log of x looks like that.
ok? e^x has an horizontal asymptote and the natural log of x has a vertical asymptote.
e^x goes through (0,1).
Log of x goes through (1,0). Ok?
30:05And log of x looks like maybe is flatting out but it does not.
Its just going up slower and slower.
Its going up.
Because remember in exponential graph it goes up very quickly because say this was 2^x, 2^10 is 1024 you get a very big number.
So log of 1024 is only 10, so now you got way out and going up 10.
30:34So the natural log graph is very flat but, it's never actually flat it's just very slowly So as log of x looks like so we can shift it and have the vertical asymptote so we can shift left and right.
So if you have y= ln(x-1), that looks the same as before except there is (2,0).
31:02And now the vertical asymptote is at 1 Log is very useful. Lots of times graphing something, like exponential, could be very messy. You cannot stay on the graph. So what you do is graph the log. Instead of graphing x vs y you graph, you make y = the log of x. The log of y. So, thats take the exponential
31:33graph and turns it into a log. So the idea is, the main value of logs of these kinds of graphs is just it is easy to graph. As you can see there is a lot of arithmetic relationship then you can get relationship of a line and that tell you that it is exponential So when you graph axis for example, you cannot really graph 10^5, 10^6, 10^7, 10^8 all on the same asymptote,
32:03all on the same axis because you are getting 10 thousand, one hundred thousand, million, very big numbers but its easy to graph 5,6,7,8, So you just graph the log of what the number is, it is much easier to fit all that in.