50 college algebra. This episode is on transformations of fundamental functions, and then some related information. If you remember back in episode three, the last episode, there were eight fundamental functions that we graph. There was the constant function, the identity function, f of x equals x, and more functions, the square root, the cube root functions. Today we're going to look at how we can transform those functions into other shapes. But to begin with, I think we should look at how we use the graphing calculator to graph some of these functions. Then we'll look at the transformations. Then we'll talk about finding x and y intercepts near the end of the hour. And finally we'll look at graphical solutions to inequalities. So let's begin with the graphing calculator. I'm using a TI-82. There are lots of graphing calculators on the market, and you may or may not have a graphing calculator, but I want to just show you how you can use some technology to graph some of the functions that we've looked at. So, for example, let's see if we turn it on here, I'm going to go to the y equal button, and I want to graph the squaring function, x squared, so I'm going to enter x, and then I'm going to square it, and I want to draw a graph of the function y1 equals x squared. But before we can draw the graph, we have to decide on where on the graph we want to look. Do we want to look up high or down low or where? So I think we should look somewhere around the origin. So I'm going to choose the window where to view it. So the first thing I see is the minimum x, negative 4 is on here. Let's just stick with negative 4, and the maximum x is positive 4. What that means is on the x-axis we'll be viewing this graph between negative 4 and plus 4 on the x-axis. Then the x scale, that says where there will be a tick mark. Where would I like for each interval denoted? So let's say every unit, so there's a 1 in that position. For the minimum y, well, instead of negative 20, I think we should pick like, how about negative 2? I don't think the squaring function ever goes negative, but let's show a little bit of the third and fourth quadrants. And for my maximum y, let's just stick with 20 like we have there. And for the scale, I think I'll choose every five units, like it says, because if I choose a tick mark at every unit from negative 2 to 20, there'll be lots of tick marks. So let's just go with every five units. Okay, the window's been chosen, the function's been designated, so let's draw the graph. And this is the parabola. Now, you notice it does go through the origin. It goes through the point 1, 1, although my scale is different on the x-axis and the y-axis, so the point 1, 1 is sort of suppressed. You remember, this is at 5, this is 10, this is 15, and this is 20 up here. So there is a point at 1, 1, and there's also a point at negative 1, 1, but it's a little difficult to see those in this particular graphic. And this is the graph of the squaring function. There's a button here called trace. And if I push trace, what that does is to highlight the middle point on the screen left to right, and that's right at the origin. And you see that the two coordinates are designated here, x equals 0 and y equals 0. Now, if I push the right-hand button, it moves that trace marker along the curve. Now, you notice it begins to rise right there. And at this moment, we're at about 1.2 for x and about 1.4 for y. If I back up just a little bit, we should be able to get close to the point 1, 1. There's 1.02 and there's 1.04. So we're pretty close to 1, 1, but now if I click it left again, I think I'll probably skip over it. Yes. Now we're at 0.94 roughly and 0.88 roughly. So we don't get to come across the point 1, 1 exactly, but I could just keep moving up the graph like this on the trace. Now you might say, well, why is it jumping from one point to another? Well, I'm going to show you how this graph was actually drawn. If I push the mode button right here, instead of highlighting connected, I'm going to go to dot. So let's click down to connected, switch over to dot, and I'm going to push enter in the lower right-hand corner of my calculator. And now let's draw the graph. And you see what happens is there are actually a sequence of little dots there. We don't have a nice continuous curve, but we have 95 dots from left to right. There are 95 dots located. And when I push connected under the mode section, what it does is it joins those with straight line segments. And what we see looks like the curve, y equals x squared or f of x equals x squared, but it's actually 95 points on that graph that are connected by straight line. So it makes it sort of a jagged thing. But there are so many points that it looks fairly smooth when we draw the graph. Now why does the calculator only plot 95 points? Why doesn't it plot all of the values from negative four to plus four? Well, that would be infinitely many points. It would require infinitely many calculations, and the calculator just can't do it. So it picks 95 values. It divides the interval negative four to four into 95 segments, sub segments, and it plots a point at every end point because it can do that and it can do it fairly quickly. And if I go back to the connected mode, I'll push enter. I'll go back to the graph again. And now we don't see the dots, we actually see an attempt to connect those points. Now you notice it does look a little jagged along here. It's not really a smooth curve. That's because these little cells or pixels are lit up, and it's difficult to arrange them into a smooth curve. So it does the best it can and lights up the little pixel closest to it. Let's take a different function. I'm going to go back to y equals, and let's take x cube. In fact, let's put x cube on a different line. So I'm going to enter x, and I want to raise it to the third power, so I'm going to push this little exponential key over here to the third power. You might say, wait a minute, you didn't push that key when you did x squared. Well, I actually have a button down here for the square because the square is such a common operation, but the cube is not quite so common, so you have to designate your entering an exponent. And I want to turn off the squaring function. So if I go up to the equal sign and push enter, you'll notice that equal sign got turned off, but the equal sign here is turned on. So when I go to graph, I'm only going to get the cubic function now. And when I draw the graph, there's the cubic function. Oh, yes, my window didn't go down very far. Let's adjust the window so we can see further down. So let's go back to the window. I'm going to change the lower bound on the window from negative 2. Let's make it negative 10. And I think I'll make the y max positive 10 so that I see an equal portion above and below the x axis. Now we draw the graph. And here's the cubic function. And you notice it comes down sort of like a parabola, but a little steeper in the middle. It's flatter than a parabola would be. And now it goes down in the third quadrant, whereas the parabola went up in the third quadrant. Now let's superimpose both of these. So if I go back to y equals, I want to turn on both functions. So I'm going to enter, push enter on the equal sign for y1. And now you'll see both of those, both of these are highlighted. So I'm going to get both graphs so they'll be superimposed. It graphs them first y1 and then y2. So we're going to graph x squared, then we'll graph x cube, because I have it set in the mode operation to graph these in sequence. So first we see the squaring function. And now we see the cubing function. Look how the cubing function goes up faster. We talked about that last time. And in the middle, it's very close. Hard to tell. But the squaring function is actually above the cubing function right here. I can zoom in if I push zoom. Zoom number two is in. Zoom number two. And I'm going to enter that. And now I'll see those two graphs a little more closely. And you see there's the cubing function. We can begin to see a space between the two functions. But I have a much smaller interval on the x-axis. This goes from, let's push trace and just see how far over it goes. If I trace over to the edge, over to the edge it looks like we're over about .87. So it goes close to plus one. And over here it goes fairly close to negative one. And class, can you tell on which of those curves is this trace moving? Is it moving on the squaring function or the cubing function? Can you tell? It might be hard to see on that screen. It's actually on the squaring function because that was the first function entered. And if I push the down button, it'll jump to the other function. Now I'm on the other function.  I'm back on the on the on the squaring function. Then I'm on the cubing function, squaring, cubing, back and forth. So if you push left and right, you actually move the tracer back and forth. If you push up or down, you jump between the two functions back and forth. So let's go over here to one half, about one half. This is getting pretty close to one half. It's .51 for x. There's .48. Let's go back to .51. The value on the cubing function is about .33. That's because one half cubed is .125, but we have a little bit more than one half. So this is a little more than .125. It's .133. If I jump to the squaring function up above, x is still at .51. But if I square that, I get about a fourth or .26. So you can see that even though it's difficult to see the difference between those two, there is a difference. And as a matter of fact, I could zoom in more and I could make that difference more pronounced, but I don't think I'll do that right now. Okay. Well, I'm going to come back to the graphing calculator in a minute to look at some variations of our fundamental graphs. But first, let's discuss those variations. Let's go over here to the marker board. Let's see. We were discussing in episode three the fact that there were some fundamental graphs with some target points that I pointed out to help us graph these. And I want to look at now how we can sort of mold or change those functions into an infinite variety of other possibilities. Let's take, for example, the squaring function. Now, the squaring function f of x equals x squared. You remember when we graphed this on the x, y coordinate system, there were some target points to plot. I'm going to keep the scale fairly big here so that this is three and this is three. And there were three target points that I would plot to graph this. There's zero, zero. There's one, one. And there's negative one, one. And then I just remember the general shape. And I draw the parabola coming down like this, coming down like this. And you remember right at the origin, it's nice and smooth and it just turns and goes back up. So it doesn't come in and make a sort of a pointy edge like the absolute value function would do. Now, what would happen if I wanted to graph this function? I think I'll call it g of x so we can keep them straight. This is the graph of f right here. Suppose I want to graph x squared plus two. Now what that does is it adds two to every y value. You see the y value here was x squared and now I've added two to every y value. So that means every y value moves up to. And so what I'm going to do is move my three target points up to. So if I label my axes, instead of going through the origin, if I raise it up to it's going to go through this point at zero two. And instead of going through one one, it's going to go through one three because I raise it up to. And if I back up one, I go up three there also. So I have my same three target points that I had here. I'm going to draw the very same shape. It's just my graph is two units higher. So I get a parabola right here. Now when I take a fundamental graph and I place it in a new position, this is referred to as a transformation. It's a transformation of the basic function f of x equals x squared. In this particular case, this is called a vertical transformation. So there are vertical transformations. That's where you move it either up or down. But we're also going to talk about horizontal transformations. We'll get to that in a moment. There are also stretches and compressions that we'll discuss. And then there are reflections where I actually flip a graph across an axis, flip it across the x axis, flip it across the y axis. So there are reflections. So vertical transformations, horizontal transformations, stretches and compressions, and then these last reflections. Before we get into all the others, let's do another vertical transformation. Class, what do you think the graph of this next function would look like? Let's call it h of x. And this is x squared minus three. How do you think I would graph that? I think it's going to look like x squared, but what change is going to be made on it, Lynne? We're going to lower it three. Just like if you add two, you raise it two units. If you subtract three, you lower it three units. Whenever you add or subtract a constant outside the function, it's a vertical transformation. So when I graph this, instead of passing through the origin like we did over here, I'm going to lower this three units. And you might say this is the quote new origin. It's the new, it's called the vertex. It's called the new vertex of the parabola. So I may refer to this as the new origin. Of course, this is always the origin back here, but I've lowered it three units. And if I go over one and up one, I get a point. And if I go back one and up one, I get a point. And now all I have to do is remember the shape of the parabola. And the curve comes down like this. And this, this is the graph of h. Okay, what if we change the fundamental function? Let's take a different one. Suppose we take the absolute value function. I'll call this capital F of x, so we don't use little f all the time. The absolute value function. Now, that function was one of our fundamental graphs. And if you remember, it has the same three target points as the cubing function or as the squaring function. And that would be at zero zero one one and negative one one. But I have to remember the shape. And this time I remember it makes this right angle with two rays coming together. There's a right angle between those. They're each at 45 degrees with the horizontal axis.  Let's see, Matt, how would I graph that?  Exactly. We're just going to take the absolute value function. We're going to raise it up four. Now, rather than just making a quick sketch of what it is, I think we can get a more accurate graph if we plot these target points. So my origin points moves to the so-called new origin at zero four. And instead of going over one and up one down here, I'll go over one and up one from that new origin. And I'll go back one and up one from the new origin. And now I remember the shape looks like this. And you know, we have a fairly accurate graph considering we only plotted three points. But we have to remember these fundamental shapes and where their target points are. Okay, these are vertical transformations. Let's take a different transformation. Let's look at a horizontal transformation. Suppose we take the function f of x equals x squared again. But I'm going to add a number inside the parentheses, x plus two squared. Now that has a different effect than if I added two on the outside. A moment ago when I put a two on the outside, it moved the graph up to. This one's going to move it sideways. Here's why. You notice f of x is a square. Therefore, this quantity can never be negative. But it could be two. Excuse me. It could be zero. And for which x will this be zero? At negative two. So let's say f of x equals zero when x is negative two. That's because if you plug in a negative two right here, you have zero squared or zero. Now that tells me that the lowest the graph can be is zero. And it will be that at x equals negative two. So if I come over here to negative two, x axis, if I come over to negative two, that must be the lowest point on the graph because we said this could never be negative because it's a square. And if it's zero there, that must be the lowest point. So that's where my new origin is. And if I go over one and up one. And if I go back one and up one, there are my three target points. I'm ready to draw my quadratic function. The quadratic function will look like this. And you see what's happened is we've actually moved it over to the left. I'm going to put a little negative two since I moved it in a negative direction. You say, well Dennis, that's easy to remember. If you add a two on the outside, you move it up. If you add a two on the inside, you move it sideways. Well here's the tricky part. If I put a plus two on the outside, plus two, I move it in a positive direction. If I put a plus two on the inside, I actually move it in a negative direction. So we have to remember that switch. Suppose I wanted to graph this function. F of x equals x minus three cubed. x minus three cubed. Well once again, I'm adding or subtracting a constant directly on the x. So that means there's going to be a horizontal transformation. Horizontal transformation. And if there's a negative here, I'm going to move it in a positive direction. So I'm going to move it over three to the right. Now of course a person might say, now I don't understand why you're moving it to the right. It's a negative three. It seems like you should move it in a negative direction. Well you remember that the original cubing function passed through the origin. So when is this going to be zero? This will be zero when x is three. Three minus three is zero. Zero cubed is zero. So when I plug in three, this function will be zero. So it goes through that point. My origin has moved over three to the right. And if I go over one and up one, I get another target point. And this is the cubing function. So if I go back one and down one, I get another target point. You remember this is an odd, this was originally an odd function. So now I draw my cubic shape, which I have to remember.  And this is the graph of that function. f of x equals x minus three cubed. Now we have a couple of graphics that help to point out the differences between vertical and horizontal translations. So let's go to the first graphic that we have coming up. Let's see now, this graphic is titled a vertical shift. It's what I called a vertical translation a moment ago. Vertical shift. So on the first line there, if y equals f of x plus c, and c is greater than zero, then you notice it takes the blue curve, y equals f of x, and it shifts it up to the red curve, y equals f of x plus c. Now if you ask me what is the blue curve, what did you graph there? That's not a function that we're familiar with. I just drew sort of a generic curve and called it y equals f of x. And then when you add c, you get the new curve, the red curve, which is higher than that, because we shifted it up c. On the bottom line, suppose we want to graph y equals f of x minus c. Again this is where c is positive. Then if you go to the right, we have the original blue curve, y equals f of x, and the red curve is below it now because we shifted it down. And the new equation is y equals f of x minus c. Okay, now there's another graphic that follows this one, let's go to. Now in the second graphic titled horizontal shifts of graphs, again I start off with the blue curve that I call y equals f of x. And if you go to the left, it says y equals f of x minus c, where c is positive. Well when you subtract a positive number, it actually moves the graph to the right. And you see that the red graph is titled y equals f of x minus c, and it moves at c units to the right. On the bottom line, if I add c to the x, y equals f of x plus c, again c is positive, then it shifts the graph to the left. And the blue graph becomes the red graph on the left. So you see that could be a little bit confusing. You have to use the opposite direction here of what you would have chosen for the vertical shift. Okay, I think we should work an example where we combine both of those. Suppose I want to graph this function. I'll call it capital G. I'll say it's a function of t, because we don't always have to have functions of x. And let's say I want to graph, let's say I want to graph the square root of t minus 2 plus 1. Well, the fundamental graph that I'm thinking of here is f of t equals the square root of t. And other than the fact that I'm using a t now, rather than an x, we know that that graph looks like this. Let's just draw a little coordinate system over here. So here's the graph. This would be the t axis. And the target points were at 00 and 11. And you remember that had the shape of half of a parabola. Let's see. It's kind of hard to draw it so small. I hope you can see this. But generally I get that shape. But my function g has two shifts or two transformations. What is the vertical transformation for my function g? Anyone? Yeah, Matt? It's going to move it up one unit. It's going to move it up one. So this plus one says move it up one, exactly. And what about this other business, Stephen? Right two units. It's going to go to the right two units. So I'm going to draw this same shape, but with those two transformations involved. So let's come over here to do that. I'll make this the t axis. And let's see, my origin is going to be moving two to the right and one up. So if I go two to the right and one up, I get a point right there. That's my new origin at 21. And my second target point, well, you know from the origin you always go over one and up one. So from this point I'll go over one and up one. I get a point right here. Those are the only two points I need to plot. And then I'll draw the shape. And let's see, remember we want to make this so it kind of turns down right there. So just do the best you can on that. And this is the function g. Let me ask you a couple of questions about that graph. You remember last time we talked about compressing the graph onto the x axis, in this case the t axis to get the domain. If I press this onto the t axis, what portion of the t axis will be covered? Sam, what would you say? I have no idea. Okay, anybody? Jenny, what would you say? It would go from positive two to infinity. Exactly. So when I press this down, it covers two to plus infinity. That's the domain of this function. And let's just write that below it here. That the domain of g would be the closed interval from two to infinity. And you know, I think that's what we would have guessed all along because we don't want the number inside here to be negative. So I have to make sure that I pick t to be two or larger. You can pick two or pick any number bigger than two, but not smaller than two. And it's reflected in the graph. What is the range of this function? Can anybody tell us? Steven? One to infinity. Yeah, because to get the range, you take the graph and you press it on to the vertical axis. And so it looks like we're going to go from here up. You might say, well, it doesn't seem to go that high, but you remember this graph keeps on going so it gets higher and higher. It'll eventually reach every number above one on the vertical axis. So the range appears to be the closed interval from one to infinity. And again, that makes sense because this is a square root plus one. In fact, this is what they call the principal square root or the positive square root. So this can't be negative. And when you add one to it, it can't be smaller than one, but it could be one or it could be larger. So the range goes from one to infinity. Okay, let's move on to another transformation. This next one is a vertical stretch. Let's go back to our basic squaring function, f of x equals x squared. But I'm going to multiply it by two so that I want to graph two x squared. So the difference is I'm not adding a two outside. I'm not adding a two directly on the x, but I'm multiplying by a two. And this has the effect of actually stretching the graph vertically. Here's why. If I begin to locate what we've been calling our target points, you notice if I plug in a zero for x, two times zero squared is zero, it still goes through the origin. But what happens if I plug in a one right here? What would be f of one? Two. It'll be two. So you see instead of going over one and up one, I go over one and up two and I get a point here. And if I plug in negative one, I also get two. So you see what's happened is the target point that was here has been stretched away from the horizontal axis. And I now get a parabola that looks thinner. But you know what? It isn't thinner, it's taller. Because all the points got further away. If there had been any points below the horizontal axis, they would have been stretched down. So if I had drawn the original quadratic function, I would have had a point there and I would have had a point there. And I would have had a parabola that would come in like this. So what I'm graphing there is the original function g of x equals x squared. So you might say it got thinner. Well I guess in a manner of speaking it, you might say it got thinner. But you know, when you get taller, you look thinner. Like you may say, Dennis, you know, you're fairly thin. Well no, it's only because I'm tall. If I were only four feet tall, still this wide, I'd look very fat. But as you get taller, then you tend to look thinner at the same time. Okay, so if I put a coefficient on here, it causes a stretch unless the coefficient's smaller than one. Let's see what happens in this next case. g of x equals one-half times the absolute value of x. My fundamental graph is the absolute value function, but I'm multiplying now by a positive factor, but smaller than one. This has the effect of compressing the graph. So let's see, if I substitute a zero for x, g at zero, the absolute value of zero, zero, so it goes through the origin. And if I substitute in a one, what is g of one going to be? g of one. It's one-half. So I only go up a half right there. And same for negative one, I only go up a half. Now this time when I draw my absolute value function, I have two rays that come together, but they don't mean it at the right angle because they've been compressed. So they come in at, well they come in in such a way that they make an angle bigger than 90 degrees now because it's been compressed.  If you put a coefficient bigger than one, it's a stretch. If you put it in a coefficient smaller than one, it's a compression. Okay, I want to put several of these transformations together before we go to the next example. What if I want to graph this function? I'll call it h of x equals three times x plus one cubed minus two. There are actually three things that are affecting the original cubic function. What's one of the changes you see in that function? See originally this was h of x equals x cubed. What three things have affected it? Lynne? Up the y-axis two. Are we going to go up two? Yes. Well, there's a nice two. Down two. Yeah, we're going to go down two. That's fine. So we're going to go down two. What other change will we make? Go left one. We're going to go left one. Yeah, if I add a plus one there, we actually move it to the left one. Okay. And what's the three going to do? Stephen. Stretch it by a factor of three vertically. Yeah. I don't know how to represent that. Maybe I'll put in a dotted line here and that's going to be stretch three. I hope you get the drift or what that's about. So if I go to my coordinate system, x-axis, y-axis, let's label or put a scale on it. Let's see. Now my origin is going to move to a new position. I have to move it down two to the left one. Okay. Let's see. Down two to the left one. So my new origin is going to be right here. And the stretch affects the other two target points. If I go over one, I go up three. And if I go back one, I go down three. That's because everything has been stretched now. And you might say, well, Dennis, shouldn't this one be stretched also? Well, you see points that were originally on the x-axis. When you stretch, it's kind of like having rubber graph paper. If you stretch the top and the bottom away from each other, the x-axis stays rigid in the middle. It's only the points off the axis to get further away. So this stretch has occurred actually now at this level, so this point doesn't move. Now I have to remember the shape of the cubic function, and that was something like this. It comes in, levels off, turned, and goes down. Now when you look at that, you say, my, that certainly is thin. Is it really thin? No, it's just tall. See, all the points got further away. So it got taller. It looks thinner. You could say that, but from this stretching point of view, it's gotten taller. Okay, let's look at another transformation. Let's see, actually we have a graphic I think maybe I'd like to show now. So let's go to the next graphic. Okay, for vertical stretching and shrinking of graphs, imagine that the blue graph that you see on each of those coordinate planes is y equals f of x. I don't know in particular what function that would be. It's just sort of a random curve. And suppose I multiply f of x times c, and c is bigger than one, then that causes the graph to stretch. And you see on the right-hand side, in that upper graph, we have the red curve that rises up higher and goes down lower than the original f of x, because it's been stretched. You notice that the origin didn't move, as well as the ends of those curves, because it seems to end along the axis. Those points stay the same, but all the other points are stretched away. On the bottom line, I multiply by a number c, but this time c is positive, but smaller than one, and this causes the graph to shrink. And if the original graph was y equals f of x, then y equals c times f of x has been compressed or it has shrunk. And so it comes up to a lower point, to a lower maximum before it comes back down. Okay, let's go to yet one more change. This is a reflection. Let me just write this name down over here. This is a reflection. And this time, I'm going to go back to the same fundamental function, f of x equals x squared. But I'm going to put a negative in front of it. A negative x squared. Well, basically what this does is it takes every y-coordinate that I got before, and if it was positive, it makes it negative, and if it were negative, it'd make it positive. Although I don't think this one ever had a negative y-coordinate to begin with. So when I graph this, my origin is still the same, because when you make the y-coordinate negative, the negative of zero is still zero. But now, well, maybe you could tell me, when I go over one, who can guess what's going to happen? If I go over one, will I go up one? Can you go over one? We're going to go down one, exactly. We're going to go down one, because you see, if I substitute in a one in here, f of one is negative times one squared, or negative one. So my function value is a negative one there, and if I plug in a negative one, I get a negative one for my y-coordinate again. I have a question. Oh, okay. On the positive x, you're still positive x-one. How did that happen? Well, let's see. So I went over to one, and I plugged in a one. Oh, so when x equals one, your y is going to be negative one. Yeah, so it's going to be negative one squared, and so we get negative one. So when you plug in a one, you get a negative one for y. So this is the point, one, negative one. That's a good question, though. Same thing over here. If you plug in a negative one, which makes it a plus one, and then you take the negative of that, so you get a negative one, so this is the point, negative one, negative one right here. Well, to kind of summarize this, what's happened is my graph used to go up, now my graph goes down, and I'll draw my parabola, like that. And you see why it's called a reflection, because it's just the mirror image of what we would have had if I had been graphing f of x equals x squared. Any other questions about that? Yeah, Matt. I had a question. Would it be a little bit more clear to write it with the parentheses so that you know it's a negative of the squared number? Oh, yeah. Well, maybe we should. Yeah, at least for now. You know, if there are no parentheses shown, then the general agreement in mathematics is that you do the square before the x. This is not a negative x quantity squared. It's the negative of x squared. Now, if you're not aware of that convention, then I could see that that would be a little confusing. But generally, when we see an exponent and a negative in front, we have to do the exponent first. That's just a general agreement in mathematics. For example, if you saw negative three squared, that would be negative nine. But if you had parentheses, negative three squared, that would be plus nine. So if no parentheses are shown, we have to decide how we're going to interpret that. And the interpretation is to do the square before you do the negative. Okay, I think we should take another example of a reflection. And let's take another one of our fundamental curves. Let's take the cube root function. Suppose we have f of x equals negative the cube root of x plus one. Well, let's see now. Just to make sure everybody remembers, the original cube root function has this shape. I'll just draw it over here. This is before we've made any transformations on it. It had three target points, and they were at zero, zero, one, one, and negative one, negative one. That's because it has an odd index. So this is symmetric about the origin. And then my graph comes in sideways like this, turns, comes around, comes back out. We could probably draw that a little bit better. And that's the graph of the cube root of x. Now, I have to make some changes in that graph. One of the changes is due to the negative, but another change is due to the plus one. The negative is going to flip it across the x-axis. We're going to get a reflection. What does the plus one do? Move it left. Very good. It's going to move it left because you see we're adding a one directly on the x. So this is not a vertical translation. That would be if I put a plus one outside. If you add a one directly to the x, it moves it horizontally in a negative direction, or to the left. So I want to draw this graph, shift it to the left one unit, and invert it. So let's see, if I shift it to the left one unit, it means my new origin, so to speak, is at negative one, zero. And from the origin, I would normally go to the right one and up one. Now I go to the right one and down one. And if I go to the left one, originally I would have gone down one. Now I go up one. And then I draw the graph. So the target points tell me how to proceed, and right here it should go, should go vertical, right there. This is a graph of capital F. Let's just try some numbers and see if that really makes sense. Let's actually plug in some values for F and see if this agrees with it. Suppose I were to find F at negative two for this function. F at negative two. Well that means put in a negative two where the x is. So I get negative the cube root of negative two plus one, which is negative the cube root of negative one. What's the cube root of negative one? Negative one. So this is negative, negative one. So I get one. So that says that negative two, I should be up at one. By golly we are. There we are right there. So we do get that point. On the other hand, what if I were to calculate F at nine? Oh I tell you what, let me change that. Let's make it F at seven. So it kind of works out in a nice manner. F at seven. Well if I plug a seven in here, I plug in a seven there, this would be minus the cube root of seven plus one. Which is minus the cube root of eight. Now the cube root of eight is two. So this is going to be negative two. And that says if I go over to seven, well seven is going to be beyond my axis the way I've shown it here. But I would go down two and that looks reasonable because here's negative two right here. Any other questions about that? Okay, let's take another change or another transformation on a function. This last one involves a stretch and a compression, but it's going to be sideways. A sideways stretch or compression. Suppose I have a little f of x and I'm going to make this change on the squaring function because we've kind of used that as our base function. Rather than multiplying by a two on the outside, suppose I multiply by a two on the inside. Now if I go to graph this function, what we're going to find is this function gets compressed and it's going to look thinner and this time it really is going to be thinner because we're actually pushing in on the sides. It didn't get taller, this time it got thinner. So let's say this is one, two, here's three, one, two, three. Let's see, if I substitute in a zero for x, I get zero for y, so it goes to the origin. But this time you notice if I go over one-half, if I plug in a one-half there, two times a half is one, what is f at one-half? One. It's one. Okay, if I go over one-half, I go up one. Now you see the way we did this before, I'd go over one and up one, but everything's been compressed. This two is making my function think my x's are coming through twice as fast as they really are. So the function thinks it has to turn up in a hurry because we're fooling it, we're giving it one-half and it thinks we gave it a one, so it turns up faster. And if I go back to minus one-half, if I substitute in a minus one-half there, that says negative one squared or one. So these points have been indented toward the middle, so I get a graph that looks like this. And it has been compressed sideways. Now from another point of view, you're probably thinking, well, wait a minute, Dennis, why don't you just square the two x and call it four x squared? Oh, well my goodness, now you're talking about the same graph, but it's a different interpretation. This is a vertical stretch of four.  So you see if you square this and write it that way, you will get the same graph, but now the graph didn't get thinner. It got taller. But it ends up being the same graph. And if I go back to negative one, I go up four. And it's the very same graph. It's just, are we going to look at these three target points or are we going to look at these three target points? We get the same graph either way. It's fine with me if you square this and think of it that way and make a vertical stretch. But I'm saying if you put it to inside, you could interpret this as a horizontal compression. Of course, the obvious question that I think is why would we want to know about a horizontal compression if we can make this a vertical stretch? I mean, why do we have to do this two ways? Well, if you take trigonometry, the Math 1060 course, you're going to see sign graphs that go up and down that sort of undulate and those will be compressed horizontally. So it's actually in Math 1060 is where you use this idea of compressions and stretches that are horizontal. Another possibility what if I have the function g of x equals the absolute value of the absolute value of 1 half x of a 1 half x? Well, I'm thinking I should be graphing an absolute value function. If I put a 2 inside, it's a compression. So if I put a 1 half inside, it's going to be a horizontal stretch. Let's see if that actually holds true. Let's see now. If I plug in a 0, I think I will get a 0 for the function value. And if I substitute in a 2, if I plug in a 2, I get the absolute value of 1, which is 1. And you see I have to go over 2 before I go up 1. And I have to go back 2 before I go up 1. And so my target points have been stretched out sideways and I get a graph that looks like this. Now, from a different point of view, of course, a person can say, well, why don't you just take the 1 half out, take its absolute value, put it out in front, and put an absolute value of x there. Well, then that will give you the same graph, but it's not a horizontal stretch. It's a vertical compression. It's going to be a vertical compression because you see at 1, I only went up a half. And if I go back 1, I go up a half. And you'd be using these three target points. So when you graph this in this form, it's a horizontal stretch. If you graph it in this form, it's a vertical compression or a shrink. Okay, in this next example that you see on the screen, the function is f of x equals 2 times x plus 1 cubed. What transformations are evident? Now, what I'd like for someone in the class to do is to say verbally what's going to happen and what's the basic function that's being altered. Okay, Jenny, in this next function, can you tell me what was the fundamental function that should be graphed or two changes that are being made to it? Okay, the function is f of x equals x cubed is the fundamental. And it looks like it's going to go to the left one. And it's going to be squished twice as tall. Okay, stretched. It'll be stretched twice as tall. Okay, let's just try graphing that up here. The function is f of x equals 2 times x plus 1 cubed. And, of course, you notice this all depends on us knowing what these fundamental graphs look like. We had that in the last episode. And, of course, knowing where the target points would originally be. So, Jenny said we're going to move this to the left one unit. So, the new origin is at negative 1. And the target points will be stretched to units. So, if I go over 1, I now go up 2. And if I go back 1, I go down 2. And so I get a taller cubic function. And if you want to say, no, it got thinner. That's okay with me. You can say it got thinner. Who's to say? But from this point of view, there was a vertical stretch, so it got taller. Let's go to the next function. The next graphic says we're given g of t equals the absolute value of t plus 3. Let's see, Matt, what's the fundamental function that we started with transformation or transformations are evident to you? The fundamental function is the absolute value of t. And it will be shifted up 3 units. It's going to be shifted up 3 units. Right. There's only one change to be made here. So, if I graph g of t equals the absolute value of t plus 3, then I call this the t axis first of all because I've called the variable t. And my target points are all going to be raised 3 units. So the origin moves to the new, what I call the new origin up here at 3 on the vertical axis. And the other target points say move to the right and up 1, move to the left and up 1. And now I have to know what the general shape of the absolute value function is and it makes this v shape characteristic of an absolute value graph. And that is the graph of g. It's not totally accurate, but none of us are going to draw a totally accurate graph. But it is quick. But if you don't plot the target points, then you tend to draw something too flat or too steep. So I think the target points sort of keep us on track for that. Okay, another topic I want to mention to you, a short one before we end this episode is to talk about intercepts.  If I were going to graph that this is basically the squaring function with two changes being made to it. Let's see, what would you say are the two changes? What's the line? The two changes, it would go left to go to the left 2 and down 9. Okay, so when I graph this save time, I'm going to quickly go to the left 2 and go down 9, and so I get a point down here. That's my new origin. And if you imagine that I go over 1 and up 1, and if I go back 1 and up 1, I realize I'm not plotting those very carefully, then I get a graph that looks sort of like this. Now, I have a different question to ask you about this graph, and that is, where does it cross the x-axis? Looks like there are two places. And where does it cross the y-axis? Now, the places where it crosses the x-axis are called the x-intercepts. Looks like we have two x-intercepts. And where it crosses the y-axis is the y-intercept. So, I'd like to locate the x-intercepts and the y-intercept. There's only one of those. And you see, anytime you cross the x-axis, we may not know the first coordinate, but the second coordinate is zero because there's no vertical elevation. Same thing over here. We don't know the first coordinate. Looks like it's probably negative, but the second coordinate is zero. So, the characteristic of an x-intercept is that y is zero. So, I'll come over to my equation and I'll make y equal to zero. You might say, well, you don't have a y. Well, the f of x and the y are interchangeable. So, I'll put a zero right here. And I have zero equals x plus 2 squared minus 9. I'd like to solve that for x. Some people might say, multiply this out and collect like terms and factor it, but I would say, take a shortcut, move the 9 to the other side, add 9 to both sides, and then take a square root on both sides. And you notice I put down both plus and minus 3 because I'm looking for every possible solution. So, one possibility, let's just put it right here, one possibility is that x plus 2 equals 3. The other possibility is that x plus 2 equals negative 3. So, we're going to get two answers for this. One of them is x equals 1. The other one is x equals negative 5. Those are the x intercepts. This must be the point 1, 0. And this must be the point negative 5, 0. Now, to get the y intercept, I do almost the same thing. If we're talking about a point on the y axis, then it's the first coordinate that's zero, but we don't know the second coordinate because we don't move horizontally. We just move down to get to that y intercept. So, this time I'm going to let x equals 0. And if I let x equals 0, I'm just going to plug into 0 right there. f of 0 equals that'll be 2 squared minus 9. 4 minus 9 is negative 5. And so, this number is negative 5 on the y axis. It's just a coincidence that I got negative 5 on the x axis as well. So, I've now found the x and the y intercepts. Anytime you have a function and you want to find its x intercepts, you let y be 0 and solve for x. When you want to find the y intercept, you let x equals 0 and solve for y in that case. Okay, let's take one more example of intercepts. You know we have time for one more example. So let's take a function g of x equals the square root of x plus 4 plus 7. Now, I want to find the intercepts for this function, but I'm not going to draw the graph. We can draw the graph afterwards if we have time. But let's see if we can do it by strictly using algebra. To find the x intercepts, I'm going to let something be 0. Who can remind us x intercepts, what's going to be 0? Y will be 0. Y is 0. It's always the other variable is 0. So, g of x and y are the same thing. So 0 equals the square root of x plus 4 plus 7. So negative 7 is the square root of x plus 4. Now, this can't happen because this is a principal square root, meaning this is the non negative square root equal to a negative 7. That's impossible. So there's no solution. What that means is there's no x intercept and when you draw the graph, this graph will not cross the x axis. To find the y intercept, I let x be 0. And if I substitute into 0, g of 0 will be the square root of 0 plus 4, plus 7. Can anyone reduce that for us? Jenny? It'll be the square root of 4 plus 7, which is 2 plus 7, which is 9. Is 9. Okay, so this graph crosses the y axis at 9. You know, I think we're running out of time so we won't be able to draw the graph to verify that. But that'll be a good activity for you to try after this episode's over. Draw the graph and see if that checks out. We'll see you next time for episode 5.


Hello everyone. I hope you guys had an excellent cloud native security day. I know I did. Unfortunately, I didn't get a chance to participate in the CTF because I was so busy reading through all the chats and listening to all the amazing talks from our fabulous speakers and helping out to everybody that reached out for assistance either through the Slack channels or through direct messages. So we're going to go ahead and wrap up the events. I'm Emily Fox. I'm the SIG Security Co-Chair and the Security Day Lead for the event. And we've got JJ who's one of my other co-chairs and Brandon Lum who is a technical lead for SIG Security. So we're going to talk a little bit more about SIG Security and how you can get involved with this awesome group of people as well as do a full day event recap and open everything up for Q&A in case you're curious about anything else going on with SIG Security or if you've got questions about anything associated with cloud native security. We've got some great stuff coming up for you. So Brandon. Thank you, Emily. Awesome. So I'm just going to go through a couple of slides. We're going to tell you a little bit more about SIG Security on top of what Emily has already shared this morning. And then we're going to go into the discussions. So awesome. What do we have for SIG Security? So Emily covered in the morning a little bit about presentations and discussions. This is something that's ongoing. As you can see, some of the topics here are something that we've seen today. For example, PASAC. There's some, there's a lot of topics that we have coming from the other working groups that we work with. For example, the policy working group. We work closely with the SIG data working group and so on. And as Emily mentioned, whenever there's a chance for community collaboration, for example, Ava dropped by from the confidential computing consortium and then we talked about the ways we could engage as a cloud native community. So if you're in a community that would like to be involved with cloud native security, or that is, you have a project that you think would be interesting to share about to come create an issue, everything that we have is managed through our GitHub. So another big thing that we have is the supply chain catalog. This was something that was created a while back by Santiago and a couple others. And there was a lot of interest in the supply chain problems and security today. We even had a breakout hallway track session on that. And I think there were a lot of interesting links that we'll put in there. I think there was even a paper. It was called The Backstabber's Knife Collection or something. Also paper is definitely something to read. It's a cool title. So this catalog is commonly maintained. So I think based on the number of resources that were posted in the channel, if you see that there is something that is missing from the catalog or we can make improvements to it, please don't hesitate to create a PR against it. All right. So usually we have in-person meetups. So this is something that hopefully next year we'll start resuming once things get back to normal. So hopefully I know Europe will be virtual, but maybe the next QCon and we'll see each other again. And we also have security assessments. This was, it's one of the co-activities that we engage in. Today we have five security assessments already done. These security assessments are a way to really assess the security posture of a CNCR project. There's a lot of interactions with the TOC and helping them provide recommendations on what is a project's security posture, as well as if it's a security project, for example, how does it really fit into the CNCF ecosystem? And the output of this project is a security document, which we, for these projects that we've already done, it is a document where if you're interested in a project or you have been looking at one of these projects, for example, OPA, and you're not sure like where do I start in terms of security considerations, the security assessment document is something that you should definitely take a look at. It really is like a good introductory document. Security assessments are also a good way to engage in the community. Usually assessments are scoped about four weeks total, where two weeks of it is the main review and assessment, and the rest is kind of putting together everything at the end and then working up a summary of what the recommendations are. So it's very nicely scoped. Usually we have four to five reviews per assessment, and so this is open to everyone. If you are someone that is new to this or would like to learn, we have opportunities to kind of shadow to kind of join the channel to see what's happening with the security assessments. And if you're someone that's experienced with this, definitely come take a look and it'll be great to have more people on board. So just a shout out to all the security reviewers we've had since we started security assessments. Thank you so much. We wouldn't have been able to do the first five assessments, which was our first security assessment master. So how do I get involved with security? Security is mainly based on our GitHub repository. So if you go there, there's meaning information, stack information, everything is governed in the GitHub repository. And so there is a new member's page that you can take a look at, tells you a little bit more about security, what we do, as well as how to add yourself as a member to the group. We have weekly meetings that you can join in our Zoom session every Wednesday. If not, what happens usually in the repo is everything is well documented in issues or pull requests. So if you actually go to the GitHub repo and you go into issues, if you've seen anything in the previous slides that you're interested in, for example, you want to present something about your project to the group or talk about the security practices within your organization, you can create an issue for a presentation and then we will start figuring out the logistics to do that you present that. Or if you have a security idea that you would like to, you think the group could benefit from, you could add a proposal or a suggestion. And if you're a project that would like to start a security assessment process, you can go ahead and create a security assessment issue. So with that, I'm going to wrap up this part of the closing. Like I mentioned multiple times before, everything is on the GitHub. So if you remember one thing, just go to the GitHub CNCF Slack security. Our meeting times are Wednesday and we have a Slack link over here of which part of the CNCF Slack. And don't forget to sign up for mailing this. All right. So Emily, back to you. That was awesome. Thank you so much, Brandon. JJ, what was one of your favorite talks for today? I know I have a list and I know Brandon's got at least one, if not two. But JJ, how about you? What was your favorite talk? Oh, JJ, I can hear you. Sorry, I'll first go to Brandon before I actually go for it. I'll have to digest my... Yeah. I think for me, I think I had a couple of favorite talks. I really liked the dynamic analysis during the S3 system calls. We were talking about EBPF and how it's kind of really evolved, how people do analysis and dynamic, even just like the runtime security stuff. I'm also like a big fan of any hardware security related things. So the past presentations and also even the certificates. I think we're talking about how we actually bootstrap the chain of trust to HSM. I think I thought that was cool. Yeah, definitely. Those ones were all really good ones. I think... So it's hard for me to just pick one, but I have to admit the UBK talk at the end of the day, right before the CTF wrap up, I thought was really cool because it highlighted a huge problem space that occurs in the community when we're trying to go cloud native because cloud native landscape is so large and security is like three miles long and leagues deep. But the nice little shopping list that they put together for how to get some of these tools to work together. We heard about that earlier in some of the other talks that you can't just pick one solution and just roll with it and you're automatically secure. You have to work on them a little bit. You have to build that defense and depth mentality across your organization because it's not just going to be runtime security. That's a concern. It's going to be all along your supply chain. It's going to be through your build. It's going to be through your pipeline, all of that. JJ, what about you? I mean, the EPP of talk was like the most fascinating one. I'd love to... There's so much about the EBPF that's actually a mystery for most people in terms of, is it secure? Is it not secure? Is it kernel mode? Is it not kernel mode? I mean, as an industry, I think there's a lot of clarification. It's super powerful as a tool to be used and super relevant for the modern day compute and modern day networking overall. Especially modern day observability that actually provides much more security at scale. I hope there is more, of course, kinds of discussions and talks that happen to ease and comfort users. There is also a lot of doubts in terms of the versions of kernel where it's applicable to. There is tremendous amount of repetition in terms of adopting EBPF. I think more of those talks will actually help ease. I think it is much more relevant for security at scale for a lot of things. That was my most favorite talk. Yeah. I think also JJ talked about the modern cloud deployments and also Emily talked about just having to juggle different aspects of security, but then even like in the shared governance lightning talk where you talked about, it's not even just different components of your security ecosystem, but when you have multiple parties involved like your cloud provider, how do you make sure that you have proper governance of that? As a company or as a cloud user, how do I ensure that even my implementation of security meets the standards? How do I do compliance and auditability if I don't own certain services? Yeah. Policy distribution at scale, computed with performance is going to be a challenge. Then the more the computers are not under your control, the more faster you want to like evaluate the security policies and it needs to happen closer to the edge. So it's going to be, that is something that actually was still fascinating and I think it should we should pay attention to that as a whole. I mean overall all the talks were phenomenal and fantastic, especially some that actually overlaid the industry was also good, but this one was probably my favorite talk. So we actually got our first question. One of our SIG security members decided to ask it. So Pushkar wants to know, what are our recommendations for security-related talks to attend for the rest of KubeCon and I can only assume that he's asking that because he's so in love with all the security talks that we had today at Cloud Native Security Day and he can't quite get enough of it as we've seen in some of the other Slack channels talking about Security Day. So Brandon, what talk are you looking forward to or would you recommend in the KubeCon schedule? I'll be honest, I was going to look through this tonight. I actually haven't had the time to look through this. I'm actually looking through it right now. All right, JJ, how about you? Yeah, give me a second. Yeah, I'm in the same camp with Brandon. I think, yeah, I'm just looking through it quickly. I think there are a few things that could be interesting. I like anything which is kind of like bypassing security mechanisms. I think the one that the Google team did, a couple of KubeCon's back on like jumping between notes, that one was cool. So there's another one called bypass Velco. So I think this could be cool. It's going to touch on a lot of the stuff that we just talked about like EBPF. It's a mystery. It's something cool, but I think this will be kind of a good insight and a nice exercise to kind of see how the TRED model of that fits into into applications as well. Yeah, for me, I spent most of yesterday furiously coming through the schedule looking at all of the talks. The bypassing Velco one definitely stood out for sure. There were a couple of other ones, though, that I thought were really interesting. Let me find that one. I mean, pretty much any talk with Liz Rice is going to be good. Yeah, man, there's just so many. And that's one of the nice things is that we're starting to see more security focused talks with or even just regular technical talks with the security bend to them, which is fabulous. And that's something that we want to see from the community. But yeah, I actually have my schedule viewable. So if you are friends with me in schedule, you can check out and see which sessions I'm going to if you're interested. Yeah, I think to stop the starboard one, starboard is something that I looked at for a bit. It's definitely a cool observability as well as kind of get a good idea of what you know, things within your cluster, what's happening there. I also I'm looking forward to the PKI the wrong way. Because that is something that although has been around is misunderstood a lot. And I think this is going to be your help solidify some of the kind of mindsets and also hopefully give me another way to see it and to see how how we can make sure we cover those grounds when we're in the PKI. I feel like no matter how many times you do it, you can bound to make mistakes. Yeah, we got another question. For compliance automation, do we have any recommendations for a specific session? I would like to point out that we had some excellent compliance check coverage today with OPA. I think there was at least three talks discussed today that mentioned OPA or talked about policy enforcement and automation of policy rules. So JJ, do you know of a particular talk or maybe even specific technology that we can point books to to learn more in that space? I mean OPA is a pretty good pretty good first step. And I think there is also a bunch around that's published closer to cloud security alliance that actually is somewhat of the intersection point of where we are. If you're looking purely for compliance, I think there are a bunch of standards based stuff that that are available that it's probably worth looking at. We do a lot of cross pollination between like a bunch of other security consortiums to ours. And I think if you were to listen to some of the security talks like in the last previous four to five blocks, I think a bunch of times compliance has come up in the conversation. And there are like few good recommendations there too. But in general, there's a good amount of resources in cloud security alliances that are going to go around and then attending security meeting for more of more of in-depth conversation would be a useful thing. In compliance is like one of those things where like when your infrastructure is moving dynamically, anything that's been defined yesterday doesn't actually apply today. And it's good to have a scriptable compliance story to go with. But what you want to know is like what, how do you verify validate and then how do you basically assert what you've said, what's happening in the system. When the infrastructure itself is changing and computes are not the thing that you actually own. To a large extent, it's an open problem. So participating in community is probably what I would say to keep ahead of what your compliance story should look like. Yeah, I think there are like two or more talks in the security track. But while we're on this, I think a couple other sessions or topics outside KubeCon. Oscar is one of them. And also just my personal way to kind of like look into the compliance, one of the things that I like doing is a lot of these products have kind of like this compliance mapping where they'll kind of say, okay, he has a NIST standard. He has where the specific controls will map onto it. We did some work as part of the SIG with the DoD. There was a spreadsheet somewhere. I think we can link it in the Slack channel later. But they kind of talked about some of the controls that Kubernetes or some kind of scanning tools are going to be. But I think compliance is something that's very specific to implementation as well. It's very specific to the type of compliance that you want to end up doing, whether it's FISMA or FedRAM or HIPAA. But usually, boy, it's not in this. So if you can map it onto some title in this 853, you should be fine. Yeah, you actually brought up a good point. And this is something that we've talked about quite a bit in the SIG is that security and compliance are often lumped together and they don't always mean the same thing. You can be more secure in some cases through implementing compliance controls. But you're not always going to be compliant when you're just implementing security out of the gate. So it's important to note that compliance rulesets and policy rulesets can help make your organization more secure when you're doing them correctly. But one does not always necessarily mean the other. And we've talked about this in a couple of previous meetings within the SIG and it's actually going to be discussed briefly in the Cloud Native Security Waypaper, which is going to go live tomorrow. So shout out to CNCF, the awesome folks there that have helped us work through the logistics about getting that document published. And it will become live in our repo after a KubeCon Cloud NativeCon is complete. The white paper also has a lot of pointers to it doesn't have a full story on compliance, but it has this amount of indications to think through in terms of compliance. So I'd highly recommend taking a look at that as well. Yeah, I think also the policy of what group that is front of NC as well has some discussions. I looked at a couple of recordings for some of the sessions and they're talking about compliance in terms of the policy framework as well. So that could be a good place to go to as well. Definitely for sure. What else did you guys like the most about Security Day? Overall, mostly because I've been so close to the Cloud Native Security White Paper and getting that wrapped up. It was really nice to see that almost every single one of the talks were entirely about Cloud Native Security end to end or at least how you manage or deal with a particular problem area in Cloud Native Security. It's I talked specifically about dynamic image security scanning and how that fits into your life cycle, which we talk about in the paper as well. Alfie and Nick talking about the importance of your data source and the benefits of understanding what the data gives you for Cloud Native Security. Kelly talked about security theater, which I think a lot of us are very familiar with and the list just goes on and on. A lot of these topics are huge problems in the area where we're trying to move security forward. There was a lot of that progression in many of the talks. What did you all think about what was the most important theme or the biggest takeaways from today? For me, actually, I feel like the cool thing was that we kind of see the talks evolving to be targeted towards a more mutual security posture of running compute. It's no longer or I'm just going to put some scanning stuff and then have the CI and have my cluster and our bank and I find we're going to be seeing a lot of more advanced problems. That's why we have Cartographer. We're looking at service mesh. We look at how do I, having service mesh side cars and authorization in Kafka, right?  We're not just talking about simple basic controls, but how do we tie the entire security picture together? I think that's what I got excited about, especially with a lot of hardware stuff. But I'm curious to hear from everyone as well, what are future topics that we can really go into? What's interesting? What are the new requirements that we're seeing as we are getting more and more books moving to the community? I mean, one of the talks related to that that reminded me of is the defender's thinking list, attacker's thinking graph. There are multiple people that talked about that as well. This isn't the first thing, but it's just fascinating as the services get more distributed, as the services are running everywhere, the defense mechanism of early days of making it perimeter based security doesn't actually really work. We have to look through the defense mechanism in a completely different way to think through it. That I would highly recommend any talk, any directionally applications, either compliance verifications or even application of authorization policies that thinks about it in a graph-based way. It's probably a directionally right thing to do. Might be too early to be honest, I think it's still a lot to be desired in terms of performance when you're evaluating in that zone, but directionally, I think that's probably a good way to think through security. Yeah, I think one of the interesting things that we got a lot of is just because you're doing containers, it doesn't necessarily mean you're secure. We saw that through the CTF and it's unfortunate that Andrew had to drop offline, but if you were online before, he did a wrap-up of the CTF and he went through all the challenges and the flags. This was our first time doing the CTF as part of Cloud Native Security Day. We wanted to do one of these for a while, and it's really awesome to see so many people, especially first-timers in security, or folks just sipping their toes into the water, to be so successful in their first container escape. Did either of you have a chance to participate in the CTF? I know I didn't. No, I didn't get to, unfortunately. Maybe next year when we can take a break from running the event. Well, we should get Andrew to give us all the access. That's right, but exactly right. The container security is super important, but container security isn't the only thing that's going on in the Cloud Native space. We had a presentation at one of the SIGs recently about serverless security from the Cloud Native Security Alliance. As of late, I've been getting a lot of newsletters from various sources about things that are going on in the security landscape. Serverless security has pretty much come up in the last 10 emails about it that I've gotten. I'm curious, do you guys think we're going to get some serverless security talks next year, maybe? That could be cool. It is a fascinating topic, serverless security, because it is the ephemerality that makes it harder to enforce the security that's actually consistent. I think it's harder, but at the same time, it's easier, because the good thing is, really, it decomposes your entire application, your entire enterprise into the smallest component. So you could technically, theoretically, have the best authorization policies ever. But then you also introduce other side of it, which is that level of fraction of compute and multi-tenancy. It's the difference between the container. Container will be more secure. Container is assumed security. The nice thing about that, and with a lot of the other talks that we got, especially from the end user stories, like with Ubiko and one of the other presentations that we had earlier, is that it kind of builds on the previous layer or the previous activities or the previous part of the lifecycle to ensure that we're getting to that more and more secure state as compute gets more and more finite and smaller. So I'm curious to see what kind of talk submissions we're going to get next year. I'm very excited about it. Do we have any other questions from folks?  At one point, we had like 60 plus people in the viewing area. Yeah, I think we have overtime a little bit. I'm not sure whether we're going to get kicked off. All right. Well, I will go ahead and wrap things up. I want to thank everybody for joining us for Cloud Native Security Day North America 2020. Thank you, Brandon. Thank you, JJ. Thank you, all of the CNCF staff that helped make this possible, as well as the Cloud Native Security Day program committee. Without all of you here attending the event and volunteering and helping out, we could not make this possible. So thank you all so much. I hope you had a wonderful day. Yeah, I have to say big kudos to the CTF team. This is the one of the Spooder CDFs I've seen run.  So shout out to Emily and shout out to Brandon for closing and then keeping it in sync. All right. Thanks, you all. All right. Thank you. Take care.


In this study, researchers investigated the prevalence of anti-beta-2 glycoprotein-i, AB2GPI, antibodies in patients with renal diseases and suspected antifospholipid syndrome, APS. They found that IgA, AB2GPI antibodies were more common than IgM and IgG isotopes in patients with renal diseases and those who tested positive for APS. Additionally, they observed that IgA, AB2GPI persisted in 22 out of 29 patients who were retested, suggesting that it may be a better marker for diagnosing APS in patients with renal diseases. This article was authored by Sabiha Anis, Ijaz Ahmed, and Rana Muzafor.


Family Theatre presents James Cagney, Gene Cagney, and today's special guest, Mr. Thomas F. O'Neill. From Hollywood, the Mutual Network and Cooperation with Family Theatre presents To the Victor, starring James Cagney and Gene Cagney. Family Theatre's only purpose is to bring to everyone's attention a practice that must become an important part of our lives if we're to win peace for ourselves, peace for our families and peace for the world. Family Theatre urges you to pray, pray together as a family. And now to our transcribed drama, To the Victor, starring James Cagney as Sam and Gene Cagney as Ellen. Well, that was quite a welcome, Charlie. Any injuries on the trip, Sam? No, nothing serious besides when boys play hard enough to win, they figure to get roughed up a little. You remember that, Charlie? How did Jim do?  Too easy going on the floor. Just won't fight. I guess we feel you've got enough fight for all of us, Sam. Well, maybe I have it that, Professor. Say, how about you and Jim dropping up to the house? Sis is fixing up some cold cuts. I'm having Gene Reynolds and some other folks over. Uh, I don't know. Jimmy tells me you're pretty tough on a man who breaks training. There'll be no training broken tonight. He'll be home in bed by 10. I'm putting Sally in charge of seeing to that.  Taking a man loving him. Well, come to think of it, Sally's probably boning up on her French for that award exam tomorrow. She's pretty set on winning it, and it's her last night to study. So Sally's finally caught the victory bug, too. But she comes by naturally, and it's a good way to be. People don't hand you prizes this world. You've got to win them. Well, see you over at the house, Charlie. Gene Reynolds, more punch? No, thank you, Miss Walters. This is fine, and a lovely party, too. Thanks. Glad you could come. Sam? Oh, just a little thanks, sis. Well, go on talking. I didn't mean to interrupt. Well, Sam was just explaining why he had the team continue to play so hard against Stonehill, even after having built up such a commanding lead in the first half. It's a case of not allowing them to let down Dean. That's the pitfall. If they stop pushing, stop driving, they may go into a slump and stay that way. But Sam, isn't a pace like that, well, isn't that rather hard on the players? In a way, yes, sis, but so's anything that's competitive. So's life, for that matter. Isn't the man who sets the pace, gets out in front and stays there? Isn't he the one you look up to? I suppose so. I've said it to every kid I've ever coached. Don't come out to play. Come out to win. Well, Sam, if this season's record has any indication, I'd say you've taught that maxim very well. Yes, we've been... Oh, look. Look, Charlie and Jim Burton just came in. Over here, Charlie. Hi, Ellen. Hi. Good evening, Dean. I think you've met my son, Jim, Dean Reynolds. Glad to see you again, young man. How do you do, sir? How are you, Jimmy? Just fine, Miss Ellen. Evening, Coach. Remember now, Jimmy, this is a very special occasion. Don't run it into the ground. I won't, Coach. I promise. Is Sally around? She's in her room studying, Jim. I promise to send you right up. Thanks, Miss Ellen. Don't stay too long, Jim. Sally's trying to win a prize or something. I won't, Daddy. Hi, honey. Jimmy. Oh, gosh, I'm glad to see you. Did you miss me? I'll say. Hey, what's wrong? You've been crying? Oh, no, it's nothing. You have, too, been crying. Now, what's wrong?  Dad's made up his mind. I can win it. And I can't. I'm just not good enough. Okay, so you can't. What's the difference? Well, it's going to make a big difference to him. A big one. A French prize? What's he care about French? It isn't that. It's winning. You're on his team. You know how he is. Yeah, but I thought he just felt that way about basketball. So did I until I started college this year. I don't know what's come over him. He's just trying to give you some of the old fighting spirit. Now, you know that. No, it's more than that. Even Ellen's noticed it. Ever since mom died and she came to stay with us, Dad's been getting more and more like this about everything. Has Ellen tried talking to him? Mm-hmm. But he still thinks she's just his kid's sister. Yes. Miss Sally. Oh, Dad, come on in. Say, Jim, there's a car stall in the driveway. Would you mind giving him a push? Oh, sure, could. I was going to leave pretty soon. Anyhow, it's almost 9.30. I'll see you tomorrow, Sal. Good luck on the exam. Thanks, Jimmy. Good night. Good night, Jim. Tell your dad I'll be right down. You bet. Night's in. Well, honey, how's the French coming? Going to sweep the field tomorrow? Well, it... Oh, gosh, Dad, I'm not very confident. Oh, don't you worry. You'll cop that prize. Just go in there to win it, and you will. Dad, it's not that easy. I'm just not the best student. Why, some of them have even lived in France. What's the matter with you, honey? You can win this thing. You can run away with it. I don't think I can. You can if you want. You're giving me the same kind of excuses I hear from lazy players. The other team was too big or too fast or too something. Well, excuses like that don't mean anything. They aren't excuses. Now, listen to me. 23 years ago, right here in this college, Charlie Burton, Professor Burton, and I played on the same basketball team. But, Dad, this isn't basketball. Oh, I don't want to beat anybody. It's not a case of what you want, honey. It's a case of how things are. I'm trying to make you see something. Charlie and I were teammates. He had a lot more natural ability than I did, but I played harder. I had a reason. They were going to pick an assistant coach from the team right after graduation, and I wanted that job. So I fought for it, and I won it. Now, what I can do, you can do. But, Dad, I don't want to win the French prize. You're lying to yourself about not wanting to win. Dad, you don't understand. I understand better than you think. You're not a child anymore.  Start acting like one. I've made a pretty good reputation myself in this college, and I expect you'd do the same. Well, I keep my marks up. You can do better than keep them up. You can be on top. This French exam is a good place to start. Well, I'm going to do the best I can. Do better than that. Win it. Oh, but, Dad... Win it! You can if you want. And I'm expecting you to. Anything wrong, sis? Sam, did you and Sally have any kind of an argument last night? Nothing serious. Why? Well, she seemed upset when she left for class this morning. I don't think she had much sleep. Probably a late studying for that French exam. More salad? No, thanks. You know, Sally can't possibly win that prize. You wait and see. To begin with, she hasn't had the background in languages. Oh, what's background got to do with it? She's taking the same course as all the others are. Oh, Sam, it's not that simple. It's not just a basketball game that she can slam through. There's more to basketball than slamming through. Then you ought to realize that effort alone won't make a good player. That's not the point. It's exactly the point. Oh, Sam, what's come over you? Sally tells me you never picked at her about her schoolwork before, but ever since she ended college last fall... It's different, Ellen. It's different. It's the most important stage in her life so far. Sam, be honest with yourself. Would Sally's showing matter as much to you if she were attending another college? Some place where you didn't coach the basketball team? I don't see what that has to do with it. I think it has a lot to do with it. Yes. I'm getting a little tired of this. You and Charlie Burton and... Well, just about everyone. Always reading me lectures on how victory conscious I am. Well...  You have been all your life. Yes. I am. And so's the alumni and the student body and the board of trustees. This victory bug I'm accused of having has earned Sally and me a nice, tidy income over the past few years. Of course it has, Sam. And no one's proud of you than Sally is. But... Winning isn't all there is to living. If you don't win, you lose. I know that. And I know that losers don't pick up many chips after the game is over. Oh, but Sally's life isn't a game. And I don't think you should have made her feel that that French exam was, either. What's wrong with it? Does a kid good to know someone's pulling for them? Well, I... I hope so. You're going right over to the field house after lunch? Yeah. I want to run the film of the Stonehill game again. Oh, I'll get it. Hello? Oh, yes, Charlie. He's here. Hold on. It's Charlie Burton. I don't know what he won't. All right, I got it, sis. Hello, Professor? Yeah? She... She what? Did you talk to her about it? Okay. Okay. I'll be right over. Thanks, Charlie. Sam, what's the matter? It's Sally. She was caught cheating in that French exam this morning. Cheating? That's what Charlie says. Oh. Her teacher came and told me. He hasn't reported it yet. Oh, Sam. Oh, I... I can't believe it. Sam, she'll be expelled. It must be a mistake. It's no mistake. She admitted it to Charlie. But what can you do? I don't know. I'm going over to Charlie's office right now. Maybe there's some way I can squelch this. Charlie, I... I just can't believe it. Are you sure? Sally admitted it, Sam. She was caught with a trot. She had a list of French irregular verbs rolled up in her hand. Who's her teacher? Mr. Dubois. Professor? No instructor. Hmm. Has he been here with Stantle long? Not too long. A little over a year. Sally, do you know? I sent her over to my place. Jim took her. Your place? Why didn't she go home? Ellen's worried to death. I called Ellen back and told her not to worry after I spoke to you. Sally should have gone home. I... I don't like to tell you this, Sam. But she was afraid to. Crazy kid. You know why she cheated in that examination. All right, all right. I wanted her to win it. But not by cheating. She knew that. Charlie. Yes? How, uh... How long with this Dubois? Last in the language department if you didn't warn him. No, Sam. I won't do that. I won't even go near it. You won't, huh? You won't. Well, I'll go near it. Everybody got a special set of high-flown ideals around here. But I don't get much ideals talk when they want a sectional championship to write up in the yearbook. Sam, take it easy. It's a different story then. I can cut all the corners I want to bring home a winner and nobody bats an eye. But let Sally get caught trying to do the same thing in a French class and it's a scandal. You never cheated to win a game in your life in your north. Suppose I had. You think that would be the issue? It would not. Is did you win or lose, Sam? That's the only question they ask around here. I put this jerkwater college on the map and I didn't do it to get shoved around by some dollar-in-hour bookworm like Dubois. Sam, nobody's trying to shove you around.  Because I can't kill it. It's got to go to the dean. Why? I'll tell you why in language that you can understand. Because I haven't spent 20 years working my way up to a professorship in this jerkwater college, as you call it, just to get it knocked from under me by a grown man who's oversold his daughter and the principal of victory at any price. Then you won't help me. I'll help you all I can, Sam, but I cannot change the rules. If this were the other way around and you asked me to help you... Suppose it were. And I ask you right now to make my son Jim a first-string forward. It's not the same thing. To see to it that he won his letter this year no matter what it did to the team. It's not the same thing. Would you do it? Is that your answer?  I am still trying to win, and I'm good at it. I'll beat you on this, Charlie. I did it 20 years ago for this coaching job. Sam, use your head. Don't you worry, I'm using it. You won't have to report this to Dean Reynolds. I'll report it. And we'll just see who stacks up biggest in this fakers. You and your Mr. Dubois or me. I wish you hadn't come here, sis. You know I've got a rule against visitors in the fieldhouse doing practice. I'm sorry, Sam. Sally and I had to see you. Sam, I'm trying to go about this the way I think Marie would have done it. All right. Well, young lady, you stood up a nice mess. Oh, Dad. Oh, come on now. That won't help. You must be so ashamed of me. No, honey, no. I understand. You wanted to win. You just tried a little too hard. That's all. I wanted you to be proud of me. I know, but look here. Would you be proud of me if I had to cheat to win games? No. Daddy, what do you think they'll do? I don't know. I'll talk to the dean. Maybe we can quiet this Dubois character down. But Sam, this isn't Mr. Dubois's fault. Who said it was? You've been talking to Charlie Burton? Yes, I called him this afternoon after Sally got home. Sam, that kind of pressure doesn't seem fair. Dubois was just doing his job. Yes, and as far as I'm concerned, if he's willing to forget about this, he can stay here and keep right on doing it. Sam, I think you're making a big mistake. Why don't you ask the dean to help instead of threatening him? I'm not threatening anybody, but I don't have to come crawling for favors either. I've done plenty for this college. Don't you think the dean knows that? He will when I get through reminding him. But don't you realize that forcing him to silence Mr. Dubois is, well, that's cheating too. Don't talk silly. What else would you call it? I call it playing tough. That's what it is. You're wrong, Sam. Don't you understand that if Dubois goes ahead and exposes this, he might wreck us? But that's not why he's doing it. Well, why do I care why he's doing it? He's doing it. That's all that counts to me. Sam! And I can stop him for the simple reason that I swing more weight around here than he does. There's no cheating involved. It's just another case of the old story, the best man wins. Oh, all right, Sam. You go ahead and win. Oh, I'm sorry, sis. It's the only way I know how to handle it. I've got an appointment over at the dean's office 6.30. You don't have to hold dinner for me. Sit down. Well, how does the team look to you today after a good night's sleep, huh? They're a good bunch of boys. Now, Dean, what I came to see you about... Before we go into that, Sam, there's something rather important I'd like your advice on. It's strictly confidential, but do you mind? Why, no. Well, a young instructor came to me this afternoon and reported that one of the students in his class, a very good student, too, had been cheating in an examination. Dean. Sam, Sam, sorry, I told you this was confidential. Now, the difficulty, according to the instructor, is that this student cheated on only one part of the examination and that even without cheating would have received an excellent mark. Not the highest in the class, mind you, but very high indeed. Well, if that's so, how come the student bothered to cheat at all? That's what bothered the instructor. He didn't know what to do about it. Sam, it's my feeling the youngster simply hasn't an overdeveloped sense of competition. I imagine you run across that sort of thing occasionally in one of your players. Yeah, once in a while. Well, I've got a hunch that youngsters like that have been taught no doubt by well-meaning parents. Nobody will ever give them anything so they'd better take what they want. It's possible. It seems rather unfair to blame a child for that, much less punish them publicly. Well, my car's in the parking lot. Can I give you a lift? No, no, thanks, Dean. Well, oh, I almost forgot, Sam. I dropped by your house this afternoon, but no one was in. I ran across something in my files. I thought that you'd be interested in. It's in your mailbox. Good night, Sam. Sam, what happened? It's going to be all right, sis. Reynolds knew all about it, but he pretended he didn't know it was about Sally. But it's all right. Yes. He got kind of cute about it, but she can stay in school. Probably get a rotten mark on French. Oh, Sam. Oh, Sam, I'm so glad you didn't have to fight about it. Yeah, that was kind of funny. I hardly had a chance to open my mouth. He just gave it to me. First time in my life that's ever happened. I guess he figured after all these years I had it coming. Now, Sam. Oh, I'm grateful, sis. I'm grateful. But, well, it's one thing for Reynolds to talk, but it's another thing to realize that I wouldn't be where I am if I weren't the kind of guy I am. All the talk in the world won't change that. How about some dinner? We waited for you. Yeah, fine. And you better go tell Sally she's out of the woods. She's in her room. You go on into the kitchen. I'll be right along. Oh, say, what's his envelope? Oh, I don't know. It was in the mailbox when we got home. Oh, yeah, yeah. The dean said something about it. I'll be right down. Mm-hmm. Well, my old letter. And what's this? Good. Oh, no. All these years and I... Sis. What's the matter? Come down here. What is it, Sam? Look at this. It's a carbon copy of a letter I got almost 24 years ago. Saying I'd been chosen assistant basketball coach. Now, I'll take a look at the other letter. Stable to the back of it. Hmm? It's dated August 4th, 24 years ago. That's two weeks earlier than this other letter I received. It starts out... Dear Mr. Reynolds, thank you for offering me the position of assistant coach on the Stanton basketball team. It's a great honor and I appreciate it. However, I'm afraid I must decline. I do not feel that I could do justice to my postgraduate work and still fulfill my duties as a coach. I very strongly recommend that you offer this post to my teammate, Samuel Walters. He is one of the finest, most conscientious players Stanton has ever produced. And most of all, he has a great fighting heart. He plays to win. Hoping you will seriously consider this recommendation, I am sincerely yours, Charles A. Burton. Charlie? I'll imagine. Hmm. And I thought my college education ended 24 years ago. Sally? Sally, I want you to do something. Yes, Dad? I want you to get on the phone and tell that easygoing boyfriend of yours that the Waters are taking the Britons out for dinner tonight. Oh, Dad. Oh, Sam. And, Sally? Yes? Tell Jim I've discovered a very old bone that I want to pick with his father. I will. Sam, you're not a winner anymore. No, I guess I'm not, sis. But you're something a lot better. Yeah? What's better than a winner? A good sport. And now, once again, the star of today's story, Jimmy Cagney. Thank you, Tony. As we all know, the purpose of family theater is to further the necessary and beneficial practice of prayer, family prayer. Just before the show, I was looking over a list of the hundreds of stars of stage and screen and radio who have contributed their time and talent to appear on family theater and in their way to urge you and your family to pray together. But the results of all their efforts would never get into your home were not for the medium of presentation which brings these plays to you each week. I'm speaking, of course, of the largest radio network in the world, the Mutual Network. And we've asked Mr. Thomas F. O'Neill, Chairman of the Board of the Mutual Broadcasting System, to be our guest this evening. I know I voiced the sentiments of all the players who have appeared on family theater over the last six years when I say thanks. Thanks to you, Mr. O'Neill, and to all your associates on the Mutual Network for making family theater possible. Thank you, Jimmy. In a way, we at Mutual feel a kinship to family theater which goes a lot deeper than a mere working arrangement. We like to think of ourselves as members of this family too. After all, Mutual and family theater are both striving to bring our listeners the same things. Wholesome entertainment plus sound guidance and information in the public interest. And in presenting family theater each week, we feel that this three-fold objective is fully attained. Anyone who has been a study listener to family theater can vouch for the consistently high and wholesome quality of its entertainment. As for the information and guidance it disseminates in the public interest, what could be of greater personal value to any American than a formula for strengthening the unity of his country at its foundation, the family? The slogan of family theater sums up this formula for unity very simply. The family that prays together stays together.  From Hollywood, family theater has brought you to the victor starring James Cagney and Gene Cagney. Mr. Thomas F. O'Neill, chairman of the board of the Mutual Broadcasting System, was our special guest. Others in our cast were Colette McMayen, Pat McGeehan, Eddie Firestone and Herb Rollinson. The script was written by John T. Kelly with music composed and conducted by Harry Zimmerman. Anne was directed and transcribed for family theater by Joseph F. Mansfield. This is Tony LaFranco expressing the wish of Family Theater that the blessings of God may be upon you and your home and inviting you to be with us next week when Family Theater will present Highway East starring Susan Hayward, Barry Sullivan and Phyllis Thakster. Join us, won't you? Family Theater is broadcast throughout the world and originates in the Hollywood studios of the world's largest network. This is the Mutual Broadcasting System.