I'm Dennis Allison in the Mathematics Department at Utah Valley State College, and this is episode eight of Math 1050 College Algebra. The first seven episodes have covered material that was the beginning of this course. Episode eight is strictly a review of material that will be on the first exam. Now let's call up the first graphic and I'll show you a few items that you need to be aware of. First of all, you cannot use a calculator on this exam. We have used calculators occasionally in class to draw graphs and we've used the graphing calculator for that. That has been supplemental and you will not be allowed to use a calculator on this test. There is a test later where you will be allowed to use a calculator but I'll advise you of that as the time comes. Also, you cannot use notes and you cannot use your text, of course. And also all of your scratch paper will need to be turned in. There will be some scratch paper that accompanies your test and you should turn all of that in with your exam when you're done. Now episode number one was a review of material from Math 1010 Intermediate Algebra and none of that review material is on this test. That was merely to sort of interface this course with the prerequisite course to give you a chance to just look over some ideas and some information that will be useful in this course. So this exam begins with episode number two and let's look at the graphic for episode number two. I have listed here several items that you need to be aware of from episode two and we'll talk about these in just a moment. I should mention that if you go to the website you'll see more items than this listed for episode two but I just picked out a few that I wanted to talk about today. So in episode number two you should know the distance formula and the midpoint formulas. There were two midpoint formulas. You should know the center radius equation of a circle and you should know how to write the general equation of a circle and you should be able to sketch the solution of a circle of a circle inequality. So let's come to the greenboard here and let's just go over some of this information. You remember there's a distance formula D equals the square root of X2 minus X1 squared plus Y2 minus Y1 squared. That's called the distance formula and then there were two midpoint formulas to find the midpoint between two points X1, Y1 and X2, Y2. So X bar is equal to X1 plus X2 over two. In other words it's the average of the X's and Y bar is equal to Y1 plus Y2 over two. Now these formulas won't be given on the test and you're expected to know these and be able to use them. So let me work a couple examples that where I have to use these formulas and then we'll look at the circle formulas. Suppose I have two points. Let's say a point P whose coordinates are seven, four and a point Q whose coordinates are one and negative six. And I would like to find the distance between those two points. So I'll choose one of these to be point number one, one of these to be point number two and I will substitute in X2 and X1 into the distance formula Y2 minus Y1 into the distance formula and the distance will be the square root of one minus seven squared plus negative six minus four squared. And that's equal to the square root of, let's see, negative six squared is 36 and negative 10 squared is 100 and so this is the square root of 136. And of course we don't have calculators available on the test so you can't approximate this but you could simplify the radical. For example, it looks to me like a four should factor out of this and so if I put a two on the outside this would be the square root of, let's say I have to divide by four. So that's gonna be three, four, 34. So two times the square root of 34 and the square root of 34 can't be reduced any further so this is the distance between the two points. If you get your answer to the square root of 136 and you don't reduce the radical I may take off a point or something for that. It wouldn't be major but I would want you to reduce the radical when you can. Now the other question related to this is what is the midpoint between these two points? So I'm gonna use the formula over here and to get x-bar to find the midpoint I'm gonna average the two x's that would be seven plus one over two is four and to get y-bar I'll average the two y's and that will be four plus negative six over two. That's negative two over two or one. So the midpoint is the point four one. If you were to plot point p, point q and the midpoint you would see that the point four one is on the line on the line segment connecting p and q and it's exactly halfway between p and q. Okay, now for the circle equations there are two fundamental equations for a circle that you'll need to know. The first one is called the center radius equation. Suppose the center is at the point hk and suppose the radius is equal to r then we can immediately say the equation of the circle is x minus h squared plus y minus k squared equals r squared. So if you were given h and k and r you could substitute that in and that's the center radius equation and you could leave your answer in this form you don't even have to multiply out the squares and that would be an acceptable answer. Now if you do multiply out all the terms you would get an x squared term from here. You'd get a y squared term. You would get a multiple of x that would come from this term when you multiply it out and you'd get a multiple of y that would come as the middle term of the square of this plus a number of constants which when added together will call c and I'll set that equal to zero. For example, h squared is gonna be a constant k squared is a constant and if you subtract off the r squared that all goes into affecting the value of c. This is called the general equation of a circle. It's also called the standard equation of a circle. Now let me just take a couple examples of problems where we would use these equations. Suppose the center of a circle is at the point four negative two. And suppose the radius of the circle is five. What's the equation of the circle? Well the center radius equation would be x minus four squared plus y minus negative two so I'll write y plus two squared equals 25. That's the equation of the circle. Now if I multiply this out and collect terms and set it equal to zero I'll have the standard equation of the circle. So this would be x squared minus eight x plus 16 plus y squared plus four y plus four equals 25. Now if I rearrange terms I'll have x squared plus y squared minus eight x plus four y and looks like we have plus 20 equals 25. But I like to set it equal to zero. So I'll write this as x squared plus y squared minus eight x plus four y minus five is zero. And if you compare that with the standard equation that I wrote here a moment ago it looks like a is negative eight. It looks like b is equal to four and c is negative five. So here is the standard equation and here is the center radius equation of a circle. I may ask you to write either one of those equations given information like this. Now suppose the information is given to you in a different form. And I think you've had some homework problems that are something like this. Suppose I told you I'm thinking I would like to know the equation of a circle that has a diameter between the origin and the point four six. Four six is right here. So in other words, if I were to draw a line segment connecting those two points that's a diameter of a circle. Now let me just sort of sketch in the circle lightly. So the question is what's the equation of that circle? Well let's see what do I need to know in order to find the equation? I need to know what is its center and what is its radius? Now its center would be the midpoint of the diameter. And we have a formula for calculating the midpoint. You remember x bar is x one plus x two over two. Well let's see x one plus x two that's four plus zero over two is two. And y bar is y one plus y two over two. And that would be six plus zero over two is three. So the center is at the point two three. Now what is the radius? Well to find the radius I need to use the distance formula. I need to find the distance from the center out to the point four six. Or if you prefer find the distance from the center to the point zero zero. I think I'll find the distance from the center to four six. I'll do that over here. The distance will be the square root of, let's see now let's write that in here two three. So that will be four minus two squared plus six minus three squared. This is the square root of two squared or four plus three squared or nine. And that's the square root of 13. So if that's the distance between these two points that's the radius. So the radius is the square root of 13. Therefore I can write the equation of the circle in center radius form. X minus two squared plus y minus three squared equals the square root of 13 squared. Well when you square the square root of 13 you get 13. And so I'll take this to be the answer because if I just say find an equation for the circle the center radius equation is fine. If I say find the standard equation of the circle you'd need to multiply this out and collect all the terms and set it equal to zero. Okay let's go to episode number two. Excuse me episode number three and we have a graphic for that. In episode number three we covered a wide variety of topics and here are a few of the things that I thought we would talk about here. Finding the domain and the range of a function based on its graph. Being able to sketch the eight fundamental functions using target points only. That is not making a table but only the two or three target points. Being able to graph piecewise functions. Knowing that there are three types of symmetry. Symmetry about the y-axis, about the x-axis and symmetry about the origin. And then knowing that some functions are called even or odd functions based on their symmetry. Well let's take a few of these items. To find the domain and range of a function let's say I have the graph of a function and suppose the function graph looks like this. I'm gonna call this function f. Now it begins at negative two and it ends at plus two. And it goes up as high as two. Looks like three is even higher and it goes down maybe to negative one. So the question I might ask you is what is the domain of this function? Now we indicated in class that if you take the graph of a function and if you press it onto the x-axis you would cover the interval from negative two to two. Those are the x's that are used and so the domain would be the closed interval from negative two to two. Because that's what's covered along the x-axis. How could I find the range from this graph? What I would do would be to compress the graph onto the y-axis and it looks like it would cover the interval from negative one to two. So this would be the closed interval from negative one to two. So I would have to know where that highest point is or at least be able to approximate it from the graph and where this lowest point is or at least be able to approximate it from the graph. So we can find the domain and the range of a function by looking at it to graph. Okay, now there were eight fundamental functions that you should know and there are target points for each one. Now just to turn the tables and make this seem a little different than the way we explained it in class I'm gonna draw the graph of the function and I want you to be thinking about what function is this. So the first function I might draw looks like this. There are target points at the origin, at one one and at negative one one and the graph looks like this. And I think I'll call this function capital F and this is the x-axis and this is the y-axis. Now we've seen that function in class, what function is it? I think it's the absolute value function. So on the exam I may draw the graph of one of our eight standard functions, eight fundamental functions and ask you to name what function it is or I may give you the function and ask you to sketch the graph. Now if you sketch the graph I want you to graph it using the three target points in this case and not making a table. For example, I don't wanna see you graphing the point two, two, three, three, four, four. That's a rather primitive way to graph it and we're trying to speed this up by knowing the target points and knowing the general shape and being able to graph it quickly. Let's take another example. What function is this? This one only has two target points and that probably gives it away already if you remember your fundamental graphs. It goes through the origin and it goes through one one and the graph of the function looks like this and I'm gonna call this function g. What function is that? It's the square root function. It's g of x equals the square root of x. It has a limited domain. The domain is the set of all numbers from zero to plus infinity including zero and the range, if I press it onto the y-axis, is all real numbers from zero up to plus infinity along the y-axis. Okay, now to ask a more straightforward question like we saw in class, what if I give you the rule of the function and ask you to sketch the graph of it? So let's say this is the function g of x equals x cubed. Well, in this case, I would mark off my scale and I would have to remember the target points for x cubed and they are zero zero, one one, and negative one, negative one. And then I have to remember the shape of the function and if you remember it comes down steeper than a parabola. It turns and it levels off at the origin rather than going back up. This one turns and goes down. The reason for that is when you cube a negative number, you get a negative result. So when you pick a negative x, you get a negative y and this is the graph of g. So you notice I didn't make a table of values, I just remembered what the target points are and I drew the graph based on that. Okay, another idea from episode number three was the notion of piecewise functions. So suppose we were given this piecewise function and this one has three pieces to it. Suppose it's equal to two whenever x is less than or equal to zero. But suppose it's equal to x whenever x is bigger than zero but less than or equal to four. And suppose it's equal to x squared. I'll tell you what, let me make that not x squared. Let me make that, let's say the square root of x whenever x is bigger than four. So whenever x is bigger than four. Now you see, this is gonna require that I graph pieces of three functions. There's the function f of x equals two. There's the function f of x equals x and there's the function f of x equals the square root of x. And somehow I'm gonna graph all of the, let's say the square root's in the wrong place. I'm gonna graph all of these in the same graph and I'm gonna just take splices of each one and put them together. So here's my axis system. There is a portion of the graph that I draw when x is less than zero. There's a portion of the graph that I draw between zero and four.  Now, for x is less than or equal to zero, I wanna draw the constant function f of x equals two. That's a horizontal graph that would go right through two and it would keep on going, except this says I only draw it for x is less than or equal to zero. So I put a solid dot there to indicate I'm including that last point. Between zero and four, I wanna graph the function f of x equals x. Now that's a 45-degree line through the origin. I have to put an open circle here because you notice this is only for x is bigger than zero and I draw a 45-degree line up to the point four, four and I'll put a solid dot there. And then after four, I graph the square root function. Now if you remember the square root function, we just drew this a moment ago, curls, turns out and goes out this way. And I believe at four, that function would have value two, four, two. I'm gonna put an open circle there and imagine that you see the square root function here and here's where it appears at that point. That's the square root function continuing, but I don't see anything in this portion because my rule said the square root function is turned on when x is bigger than four. So if you put this all together, this is the graph of f. It's called a piecewise function. These functions are very useful in applications of mathematics because in the real world, things aren't as simple as having one function that's used all the time, but on occasion you use one function, on occasion you use another function. Okay, let's see. Something else in this episode I believe was intercepts. Let me just check my list here. Yes, intercepts, no? Oh, symmetry, symmetry was the other thing. Here's a typical question that I could ask you on the exam. By the way, let me just say once more, I think I mentioned this at the beginning. If you look on the website, you will see this list of topics for each episode and then a few more things. I've picked out only a few of the items that are on the website to discuss in this one hour with you. So for a more extensive list, look on the website and you'll see this and more. Now, suppose I have a graph that looks like this and I tell you that this is only half of the graph, there's another half on the other side and the graph is symmetric about the y-axis. So let's say f is symmetric about the y-axis. So I would ask you to complete the graph and draw the other half of it. Well, when we say symmetric about the y-axis, that means that if I take it and just flip it over, it has the mirror image on the other side. So it's gonna come down to a point right about here and it's gonna go up to a peak right about there and it's gonna have an x-intercept right about there. I'm just sort of estimating it, obviously. In fact, that point's probably a little bit too far out. I'll move it back a little bit. So the rest of the graph should look something like this and you may say, well, Dennis, that's not an exact copy of the right-hand half. Well, it's close enough. If you can draw it this well, I would certainly accept it. So I just want to see that you understand what it means for a graph to be symmetric about the y-axis. You may remember that the function f, in this case, is said to be an even function. An even function is a function whose graph is symmetric about the y-axis. Okay, let's take another example. This time, what if I draw half of the graph and I'll draw that much of it. I'm gonna call this function g and what if you're told that g, or the graph of g, is symmetric about the origin. About the origin. Now that has special meaning and it has important applications to later courses in mathematics.  So what's gonna happen is this function is going to turn and let's see, it's gonna turn and it's gonna go up and it's gonna go down sort of like that. Now, if you have a little trouble drawing that, I would say plot the lowest point here, go through the origin and there should be another point just like it on the other side. So put a dot there. And if there's an x-intercept here, go through the origin to the other side and put a point at the same distance away. And let's just pick a random point up here. If I go through the origin, the same distance on the other side, I should get a point right about there. And now, if I go back and connect these points, I have the other half of this graph that's symmetric about the origin. Okay, so you notice this course deals a lot with graphing. I wouldn't say that every episode is gonna have as much graphing in it as we've seen so far, but that's one of the basic features of college algebra is that it discusses graphs to a great extent. Just back to this function one more time, we might say that G is said to be a blank function. And I'm wondering if you can think, those of you at home, if you can think of what kind of a function this is, the answer is it's an odd function. An odd function is a function whose graph is symmetric about the origin. Okay. Now, what if I give you the rule of a function and I ask you whether it is symmetric about the X or the Y axis or neither one? Let's take this function. F of X equals three X squared plus the absolute value of X. Now, frankly, I have no idea what the graph of that looks like, but I can tell if it's an even or odd function or neither one, if I run a simple test. If you remember, for an even function, F of negative X should equal F of X. And if it's an odd function, F of negative X should be the negative of F of X. So I should check to see if this function satisfies either one of these rules. So I'm going to calculate F of negative X and see if I come up with either of these answers. F of negative X is, I'm going to put in a negative X here and a negative X here. That'll be three times negative X squared plus the absolute value of negative X. Well, you know, when I square negative X, that's no different than squaring X. So I could just call that three X squared. And when I take the absolute value of a negative number, that's the same thing as taking the absolute value of a positive number, so I can call that the absolute value of X. But wait a minute, that's the same thing as F of X. So what I have here is that F of negative X equals F of X and therefore this is an even function. Now you might say, well, what is the significance of that? Well, it means that if I draw the graph of this function on the right-hand side of the y-axis, whatever it might look like, if I just flip it over, I'm going to get that same graph or the mirror image of that graph on the other side. In this course, I can't go into all the applications this has, but this becomes a very useful shortcut knowing if a function is even or odd in the context of certain applications of mathematics. If you find out that F of negative X does not equal F of X and that F of negative X does not equal the negative of F of X, then the function is neither even or odd and there are lots of functions that are neither even nor odd and those functions have no symmetry about the y-axis or about the origin. It's just like some numbers are not even or odd. Four is even, seven is odd, but 3.2, 3.2 is a decimal that's not even or odd, so there are lots of numbers that aren't even or odd. Okay, let's go to episode number four and look at a few topics for that episode. In episode number four, I think probably the biggest thing we looked at was transformations of the fundamental functions. Now what I mean by transformations is vertical shifts, horizontal shifts, stretches and compressions vertically and horizontally, and then there were the reflections where you flip a graph upside down. You should be able to name the rule of a function based on its graph, that is a fundamental function or a transformation of a fundamental function based on its graph, and finally you should be able to find intercepts of a curve. Now, let's just look at some examples of this. Suppose you were to be asked on this exam a question like this and you know frankly, I bet you will. Not this example, but something like it. Suppose I wanna graph f of x equals three x squared minus two. Now we don't wanna make a table to graph this, we wanna use target points and what we know about fundamental graphs and transformations. The negative two tells me this graph will be shifted down to, and the three tells me there's gonna be a stretch. Now how can I say that? I'll just say stretch below that. So there's a stretch and there's a vertical transformation downward. So I'm gonna locate my target points for the fundamental function. And of course a good question is, what is the fundamental function? Well the fundamental function, let me kind of put it in a cloud like I'm trying to think of the fundamental function. The fundamental function before all these changes were made, before I shifted it down and before I stretched it, it was just x squared. And that has a graph that's a parabola and it has three target points, zero, zero, one, one, and negative one, one. Now I'm gonna shift that down two units, one, two. And then from this, what we've been calling the new origin, I'm gonna stretch the graph vertically. So if I go over one and up three, one, two, three, I get a point right there. And then if I go to the left one, I go up three, one, two, three, I get a point right here. So now I'm gonna draw this parabola, but it's a taller parabola is the way we described it. It may look thinner, but actually it's taller because we've stretched it. And this is the graph of F. It looks thinner than a typical parabola or the fundamental parabola over here would have looked. But that's just because we've stretched it three units. Okay, now you've seen me do lots of these examples in class, so let me change this. I'm gonna draw the graph and I wanna see if you can think of a rule like this that would represent that graph. Okay, suppose the graph, let's see, let me try another marker here. That one seems to be kind of fading out. Here we go. Okay, the question is, what function am I graphing here? What familiar fundamental function or transformation of it am I graphing? This function looks like this. I'm gonna plot a point here. I'm gonna plot a point here and plot a point here. And this function comes down like this. It levels off there and it turns and it goes down here. Suddenly it got really wide there at the bottom. And let's say this is the x-axis, y-axis. And let's say I call this function H. So my rule is gonna say H of x equals something. Okay, well, first of all, we have to decide what's the fundamental graph that we're looking at before it was transformed. Well, you know it looks to me like it's the cubic function. If you remember, the cubic function had three target points and the cubic function looks like this. And that's sort of the general shape of this one. Now, I would say it's been flipped over for one thing. It's been moved to the right one unit. And has it been stretched? I don't think so. When I went over one, I just went down one. And when I went back one, I went up one. It's been flipped over, but it hasn't been stretched. So now, how do I write that in the form of a rule for a function? Well, if I move it to the right one, that means I need to put x minus one. I need to subtract one directly from the x. And it's the cubing function. So I'll put a cube on it. And it's been flipped over.  And that's all you have to write. Negative the quantity x minus one cubed. Let's do another one like that. This time, I'll draw a function that looks like this. And this function has a point right here. And it goes down through the origin. And it goes down through, this is two right here. And this is three up here, that point is three. And I'm gonna call this function a little f. Oh, by the way, let's say we call this the t-axis. And this the y-axis. So that means I'm looking for a function that I'll call f of t. F of t. Well, I think you would probably agree this looks like the graph of an absolute value function but it's been flipped over. Just like that cubic function just before this was flipped over. So I'm thinking this is basically gonna be f of t equals the absolute value of t with a few bells and whistles added to it to make those changes. Let's see, it looks like we've moved to the right one. And it looks like we've moved up too. If I move to the right one, just like in the last example, that's gonna be the absolute value of t minus one. That moves the absolute value function to the right one. And I wanna move it up too. Okay, oh, let's see. By the way, this has been inverted so I'm gonna put a minus on here. And not only has it been inverted, it's been stretched. Look, if I go over one, I go down two to get to this point. So it's been inverted and stretched. I need to put a negative two on there. And then finally, it's been raised up two units. I'm gonna put a plus two out here. So I don't multiply the whole thing by two, but I multiply this portion by the negative two and then I add the two on at the end right there. Okay, so this is a function, or this is a rule of a function whose graph would look like this. It's the only one we can think of that would look like this. So that's the rule that we're looking for. You should be prepared to work problems like this on the exam. Okay, another notion that we had in the same episode was to find intercepts. Let me just mention this briefly so we can move on to episode five. If you're looking for x-intercepts, the way you do that, the way you find x-intercepts is you let y equals zero. If you're looking for y-intercepts, then you let x equals zero. And what I mean by this, if you're looking for x-intercepts, you let y be zero and you get an equation and you solve for x and any solutions becomes an x-intercept. If you're looking for y-intercepts, you let x be zero and any solutions become a y-intercept. Okay, let's go to episode number five and look at a few items for episode number five. In this episode, we talked about how functions are used as models of real-life applications. This is one of the fundamental ideas about all of mathematics and why you take college algebra, for example, in college, is because many other majors use material in college algebra in particular functions used as models. Then you should be able to interpret the graph of a function when it's used as a model and you should know that there are three types of variation. There's direct variation, inverse variation, and then there are compound or combinations of those compound variations. Okay, functions used as models. Let's take an example. Suppose I had a roll of fencing and I were gonna make a garden out of it, but a little different than the garden that you saw me make in episode number five. This time, I wanna make a rectangular garden and I wanna divide it down the middle in two places so that I divide it up into three pins like this. And let's say that I have 300 feet of fencing. Well, with 300 feet of fencing, I'm looking for a function that would tell me what is the area of this garden as a function of X? And you might say, well, what does X represent? Well, let's say X represents the length of this side over here. So if that's what X is, and if this is a rectangle, and if I've divided it in the middle with some of this fencing into two more sections, so I get a total of three sections altogether, then I would like to find a function that represents the area of that pin. I'm gonna call the function A because it represents area, so generally you name a function with a letter that reminds you what it's supposed to stand for. Well, I'm thinking this would have to be X, this is X, and this is X. So how much fencing is left over for these two sides? Well, I guess there would be 300 minus 4X left over and I have to divide that in half and put half above and half below, so I have to divide that in half and that gives me 150 minus 2X. So I put 150 minus 2X here and I put 150 minus 2X here. Now if you were to add up X and X and X and X and 150 minus 2X and 150 minus 2X, if you add those all up, you'll get exactly 300 because that represents all the fencing that I had to begin with. Now what is the area of this garden? Well, if I look at the big rectangle, it should be length times width. The length is 150 minus 2X and the width is X and that's the total area inside the rectangle. So I would say the area is 150X minus 2X squared and that would be measured in square feet. You don't have to multiply this out. If you wanna leave your area function as this product, that's fine with me, or you could multiply it out like this and by the way, you don't have to even put the square feet on the end. I just did it because it seemed to be appropriate to put units on the area function but this is a function that models the area of that pin. For example, what if I told you that X, that X here was going to be let's say 10 feet? 10 feet, 10 feet, 10 feet, 10 feet. What's the area of the entire garden? If I just plug in 10 into this, I'll have an answer. Plugging in 10 here, I get 1500. Plugging in 10 here, I get minus 200. So I get 1300 square feet. So that would be the area of the pin if this is chosen to be 10. Okay, along a different line, you may remember that we talked about a projectile problem in which we throw an object off the roof of a building and we find out how high the object goes and what time it hits the ground and so forth. That application used a formula that I would give you on the exam if I should ask you about this. This is a formula that was first discovered by Galileo around 1600 and the formula is minus 16 T squared plus V sub zero T plus H sub zero. Let me just draw a little illustration of this problem. Suppose we're standing on top of a building and the building is H sub zero feet tall. This is given in feet and we throw an object straight up into the air and it comes back down down the side of the building and eventually it's gonna hit the ground. And let's say that initially we gave it a speed, the initial speed, well actually velocity, the initial velocity that we gave it was V sub zero. The difference between speed and velocity is speed can't be negative, velocity can't be negative and if V naught were a negative number what that would mean is we actually throw the object straight down rather than straight up. So V naught could actually be negative in this problem. So H sub zero or H naught is the height of the building. V sub zero or V naught is the initial velocity and the minus 16 is always minus 16 on the earth. This has to do with the acceleration of gravity. This number would not change unless we went to outer space if you went to another planet where the acceleration of gravity were different, this number would change but for us it's just gonna be a constant. And this formula calculates the height of the object in feet. Okay, so if I were to ask you a problem like the example that we worked in class about a projectile being thrown off a building I would give you this formula so that's not one that you have to create. Now graphs of functions that represent real life situations. Let's take an example here, similar to some we did in class. In class we talked about the height of the grass as if you mow the lawn every Wednesday and you remember the height of the grass was sort of sloping upward, it would drop back down, slope upward, drop back down. Another problem we had was the traveling salesman as he travels away from home and then he eventually returns to home at the end of the day and you can read the graph to see roughly how far away from home he is at any given time. Let's take another problem. Suppose this time we have an oven. Let's say this is an oven here and here's the oven door and I'm gonna put a baked potato in the oven. So this is the baked potato. And so we leave the baked potato in the oven for quite a long time, it's really hot and we bring it out into the room and we allow it to begin to cool. I wanna draw a graph that represents the temperature of the baked potato over time once I bring it out into the room. So the graph would look like this. This is T measured in hours, time measured in hours. And this is capital T for temperature measured in degrees Fahrenheit. And I'm not gonna draw a negative T axis because this problem begins at the moment that I take the potato out of the oven and I take it out of the oven at the moment T equals zero. So that's when the problem begins. And I'm not gonna draw a negative T axis because that would have it going G well below freezing. I don't think that's gonna happen to the potato. It's gonna be very hot and it's gonna cool. Now how do you think it would cool off? What would the graph look like? Well I think it would begin at a rather high temperature. And let's say when I first take it out of the oven let's say it's at 220 degrees Fahrenheit. Now you know if you bake a potato for let's say an hour in a 350 degree oven that potato will not be 350 degrees. It'll be cooler, a little bit cooler inside. But it's cool, it's hot enough to cook it. So it starts off at 220 degrees. And then the temperature begins to cool off.   And it begins to level off at a line that I'll draw along here. And it gets cooler and cooler and cooler. But it never drops below the dotted line. And you know what the dotted line represents room temperature. Now let's say the room temperature is 75 degrees all around the oven. So this is 75 right here. And the potato approaches 75 degrees but it never theoretically, it never quite gets to 75 degrees. So here is a graph that represents a real life problem. And as I read this graph I can see what's happening to the potato. But I drew the graph based on some assumptions that I sort of know about the potato ahead of time. Okay. Let's see, let's go to episode six and look at some things we wanna review in episode six for this exam. You should know that there are two ways to find the vertex of a quadratic function. I didn't have room to put quadratic function on there. We'll talk about this in just a moment. You should be able to solve applications of quadratic functions. You should know about the algebra of functions. How do you add, subtract, multiply and divide functions. And finally, you should know how to compute the composition of two functions. Okay, let's consider those topics quickly here as the time remains. Suppose I have a quadratic function like this. f of x equals three x squared plus 12 x minus, minus 11. And this is what we refer to as a quadratic function because the expression on the right hand side is a quadratic. You're familiar with quadratic equations. So this is a quadratic function. And let me just ask you a question to be thinking about at home. What does the graph of a quadratic function look like? Well, it looks like a parabola.  And I can tell that because there's a positive on the square term. Now, where is the vertex of this parabola? Well, there are two ways to find the vertex. One of them is to complete the square. The other one is to use a formula. If I complete the square, I would do it this way. And if you want to use this method on the test, it's perfectly fine with me. I will not specify whether you should use the formula or complete the square. So I'm going to factor out the three and put the minus 11 outside. So I factor the three out of the x squared and I factor the three out of the 12x. And then I complete the square. Now, what number should I add on here to make that a trinomial square? The rule is once this coefficient is a one, I take half this number and square it. Half of four is two. Two squared is four. So I have to add on a four. Now you might say, is that legal to just add a four in the middle of a problem? Well, it is if you balance it. You see, I've actually put in three fours, three times four. So I've really added on 12. So over here, I have to subtract 12 to balance it. So this says f of x equals three times x plus two squared minus 23, wow. Now this tells me that basically I started off with the quadratic function f of x equals x squared. I shifted it two to the left and down 23. Well, if you go two to the left and down 23, that means the vertex is at negative two minus 23. That's where the vertex is for this function. But you know, there's another way to do this and you're perfectly welcome to use the alternative method on a test. And let me just write it over here. I'm gonna rewrite my function, f of x equals three x squared plus 12 x minus 11. There's a formula that will calculate the vertex. The x coordinate of the vertex is found by taking negative b over two a. I've used this several times in the episodes that we've watched. And that would be negative 12 over, let's see, two times a is six. And so x is negative two. Hey, look over here, x is negative two. That's exactly right. And to get the y coordinate, what I do is substitute in negative two into the function to calculate the y. That'll be three times negative two squared plus 12 times negative two minus 11. And that's four times three is 12. 12 minus 24 minus 11. Let's see, now that's negative 12 minus 11 and that's minus 23, there you have it. You can use either one of these methods if I ask you to find the vertex of a quadratic function. That's a fairly standard question in college algebra, so you'd certainly wanna be prepared to do that. Okay, another question is applications of quadratic functions. Let me just go back to that Galileo problem and remind you of this. Suppose I told you that I were gonna launch a projectile from the top of a building and its initial speed, initial velocity was, well, let's see, let's say 60 feet per second. 60 feet per second, that's the initial speed going upward. Now of course it starts slowing down, slowing down and it comes back down to the ground. And let's say that the building is 100 feet tall. So the question is how high, how high will this projectile go? Well, I'm thinking that I should use our function for the height of a projectile and that will be minus 16 T squared plus 60 T plus 100. I'm just substituting V naught and H naught into this problem. Now to find out how high it goes, I wanna find the vertex and I'm gonna find it as quickly as I can. So I'm gonna use the formula that T equals negative B over 2A. And that's gonna be negative 60 over negative 32. Negative 32, because I'm doubling negative 16. Now let's see, I can divide top and bottom by negative four and I get 15 over eight, 15 over eight. Now that's gonna be in seconds. What that says is the projectile goes up to a maximum of height and it reaches it after almost two seconds, 15 eighths of a second. How high will it go? Well, if I substitute in 15 eighths and calculate that answer, if I just plug in that number here, here, add that all up, I'll find out how high the object goes. I'm not gonna work that out because I'd like to move on to some other things before we run out of time. Okay, let's go to episode seven. And the things you'll wanna know from episode seven, which we just talked about the last time, we met here in this course, you wanna know about one-to-one functions and the horizontal line tests, those are related to one another. You wanna know about the inverse function property, which says if you take the composition of F and F inverse of X, then you get X as a result. And that's if you take the composition in either order. You should know two ways of determining and then inverse functions and that is by taking the composition of two functions to see if they're inverses or to calculate the inverse algebraically. And finally, you should be able to sketch the graph of the inverse function of a function when you're given the graph of the original function. Okay, let's just go through a couple of those ideas quickly here. Suppose, for example, that I draw a graph of a function and let's say it looks like, it looks like this.  That's pretty weird-looking function, but I'll call that function F. So let me ask you about this. Does this function pass the horizontal line test? And I think it does because if I draw horizontal lines through the graph, I think they intersect the graph at only one point. Therefore, this is a one-to-one function. What's significant about that? Well, it has an inverse. So there is an inverse function that I should be able to graph on the basis of this. And the way I find the inverse function is I draw a 45-degree line in here and I just imagine flipping this graph across the 45-degree line. So I would get a point right about here. I would get a point right about here. It looks like that portion would be straight right there. And then this portion, I'm guessing, would look about like that. Now, this new graph that I'm drawing in is the graph of F inverse. This is just one of many things you need to know about inverse functions and one-to-one functions. Okay, just to take one more idea about inverse functions, you should be able to calculate an inverse, the inverse of a one-to-one function by using the algebraic method. Suppose F of X was equal to six X plus three. And I'm wondering what is the inverse function for that? What's the inverse function? Now, there is an inverse function because if I graph F of X equals six X plus three, this is a diagonal line and it passes the horizontal line test, so it has an inverse. Now, what is the inverse? I'm going to replace F of X with Y. I'm going to solve for X. I'll have to divide by six here. So I divide by six. And then if I turn it around, X is equal to Y minus three over six. Now, if I interchange the X and the Y, Y is equal to X minus three over six and therefore F inverse of X is X minus three over six. Now, this was a rather brief example of using the algebraic method to find an inverse function. Okay, let me just go over a few items that I think you need to pay particular attention to as you study for this test. This is not a multiple choice exam, but you will need to show your work to solve these problems. So there'll be some space provided after each problem for you to fill in the details. You don't want to just give an answer when there's obviously some work to be done, so you should show that work on your test and then it's for your benefit, you get partial credit for the any work that you do that's correct. Whereas if you just give an answer, you see that your answer might be wrong and I couldn't give you any partial credit for that. You should know that you can't use a calculator, that you can't use notes and of course you can't use your textbook. So you wanna know that as you go into the test and so I wish you the best of luck on this exam and I hope you do well and I'll see you next time for episode number nine.


It is time for the shams scoops while you get paid the big bucks shams. What's the latest on Mitchell Robinson? This is an absolutely crushing development guys He is now projected to miss the rest of the season He had ankle surgery last year the Knicks initially said it would only be eight to ten weeks But I'm told the Knicks have applied for disabled player exception with seven eight point seven point eight million dollars Which is is that that that means that they're projecting Mitchell Robinson is done for the year NBA doctors are now going to conduct a review Of his of his ankle of the injury And and the Knicks Are in a tough spot now because Jericho sins is out for at least one to two weeks And now Mitchell Robinson could be done for the entire season Uh, I expect they will be active in the in the marketplace potentially to look at centers That is a tough tough blow. Lou. They sit in the five seed kind of right in the middle of the pack How hard is this for them moving forward who's been so critical to the team? Yeah, this is going to be tough like shams just mentioned not only Mitchell Robinson, but but the backup is also hurt So size-wise this is going to be a big blow to this basketball team moving forward. And so Um, you know, like shams just mentioned they're going to have to be active in the market and see if they can plug in Some of these holes, but size-wise this is going to be an issue for the for the new york Knicks Shoms ben Simmons has been out since early november with some nerve Irritation, but can you tell us the latest on him and can we expect him back? Anytime soon Ben Simmons has resumed some on-court activity individual work, but uh, the net said yesterday He will be reevaluated another two weeks.  How do you approach it if you're ben Simmons right now? And you've dealt with this for multiple years I mean man, it's tough. There's nothing worse than injuries and it's hard It's hard to keep, you know your mental strength You know where it needs to be to continue to go every single day and rehab and practice and stay in shape And you can't emulate playing in a game. You see john moran the other night as good as exposed to be still was He was still guest and he still wasn't in NBA game shape. So for ben Simmons, you know, it's been through a lot I don't know the exact issue what it's been whether it's been his back his nerve, you know His mental health is kind of a combination of everything But just feels like in this situation when it rains and pours man it continues to get worse But hopefully he's got great people around him. Hopefully he's got the support system he's working his tail off because The guy was a special player the guy's 6 10 point guard who had an unbelievable IQ his vision the way he passed the ball So it's crazy kind of what's happened to him, but hopefully he can get fully healthy soon Um handle whatever going on, you know with his mental with his mental issues and kind of get back on the court Because he's a heck of a player when he's talented and fully ready to go Uh, and same thing here with with ricky rubio. He hasn't played this season while taking care, you know of his own mental health What can you tell us about him? Yeah, this is a guy you both have played against and i'm told the cavaliers believe ricky rubio has played his last NBA game 12 seasons We wish him nothing but the best as he spends time in spain from what i'm told hasn't done anything basketball wise In months, uh, isn't really preparing to return to action anytime soon As he deals with with with mental health issues Besides have begun discussions i'm told on a potential parting of ways that could be a buyout That could be a wave that could be a trade is a team that's dealt with a lot of injuries There's garland ebb and mobily both underwent surgeries Garland for a fractured jaw mobily for his Knee there's a team that needs another body. So we'll see how they free up that roster spot That's now currently being held by ricky rubio


Edge computing comes with the challenge of scale and so when we think about edge computing it's really about scaling your people it's about scaling your operations and how do you maintain centralized control and visibility and manageability in a way that allows the business to scale as well. So imagine in a manufacturing environment you've got a lot of different plants located perhaps all over the country perhaps you're building bikes and you're making sure that the wheels are perfectly circular if for any reason there is drift in that manufacturing process you can quickly process the fact that that is happening and stop manufacturing and so what that does to a business is they don't have to waste a lot of material it's really about helping solve things at the place where that thing is happening. Look at this you're not going to believe this manufacturing error. Bicycles around the world are spontaneously collapsing a screwed use in production was manufactured too small. At Red Hat we did this really cool video where we try to bring this use case to life a little bit more.  It is put in a quite satirical type of way but it could happen and these are the types of things that we want to address and help solve for with Red Hat OpenShift. At Red Hat we have what are called Red Hat Edge validated patterns and what they do is they architect a whole edge stack that then a technology partner can add their specific capabilities to to address specific use cases so these are validated they're tested they are lifecycle managed which means that if a product within the validated pattern gets updated the validated pattern itself gets updated so what this does to the business is it helps minimize risk it helps you get from zero to POC as we say a lot faster and ultimately helps you build an edge architecture that is specific to you and is not something that is generic and it's something that you can use as an organization.


So happy to be here with Sonya, one of my favorite people in the world and Sonya you're going to say you say that everybody. No, it's true. It's true. So, yeah, Sonya is in my one of my group coaching programs right now and so we're going to be talking with Sonya several times this year and hopefully you'll kind of see the progress of growth and different tips that you're going to learn from from from Sonya's progress each time. So without further ado, Sonya welcome and let's start with your intro. I know there's a story there that you might want to tell and but let's let's let's let's go there first. Okay, I'll start with the y'all start with the actual intro before I tear to shreds. I provide magical support support services to help people find direction or purpose. Boom, I said it. Nice. I can tear into it. I these these. This idea of introducing yourself with this really tiny little thing. It just feels so calculated and cold to me that I keep trying to find something that sounds authentic and real. And I feel like so much of the world has fallen into this place of like well I got to make up.   And you know, I just feel like there's such a, it's, it's been a thing for me for a long time and I think this is part of why your stuff appeals to me so much is just coming through authentically right like when I was in the business world and all these business languages started coming up in tech of people and it's like you know we already have words for that. We already have ways to say those things. Why are you making up new language to make yourself sound more impressive can we just talk about it authentically so that everybody actually understands what you're saying. And that's where most of my struggle is with trying to come up with these things is, is like, I just want to talk about it I don't want to have a concise perfect little container to put it in, but I recognize that I need that little to exist out in the, you know, in little blurbs that only give you so many letters. As a first step into your work.  But the front door has to have some recognition. Oh, that's a front door I can enter there, I guess. And it's like everybody I feel like everybody's trying to make their front door look like the perfect sliding door on a on a business, you know, a very pretty business building a good analogy. Right. But like, I want the Hobbit door. Yeah, you know where there's vines and stuff, or like I want the sci fi door where it's like you're not exactly sure where the door is but you're going to find it and when you do you're really good. Yeah, yeah, yeah. That's that's such a great analogy because the front door to our business is made for and meant for our ideal clients, and it's also meant to repel maybe too strong. Yeah, but at least not repels good. Yeah, let the less ideal clients just walk on by. Yeah, you know, and and if your ideal client has no interest in walking into the big shiny business building like why are you trying to make the big shiny business building door, right, like, yeah. Yeah.  Right, exactly. Yeah. Well, so, so we're going to talk, I'm going to talk with you on this, these interviews several times during the during this year and so hopefully the audience can follow along your journey. Please feel free to look at the links. I'm just talking to the audience here. Look at the links below this video or this show notes, just to get, you know, access to Sony's content and kind of follow follow but we're going to talk about your current, you know, learnings and your progress and your priorities right now. So, why don't we start with, well, I know we're just we're just getting started in the group coaching program this year but you've been kind of around my content for a little while. I'd love to hear any progress you want to share what you've done thus far. Yeah, well the thing that stood out to me as I was kind of reviewing my life yesterday and your career is like, I, so what I'm noticing right now is there's really a push. There's a there's a conflict between wanting to, like, like how am I going to structure my business what's the structure that supports a sustaining business because I've had a lot of ups and downs I've had a lot of movement that you know starts that that get a little bit, you know, and there's a huge long trail of the usual like oh my business was me finding myself at first right there's that that whole history feels like it's recently wrapped up. Thank God I mean it never stops but also, you know I feel like I'm pulling up a little on that one. But right now the real push is like, especially being in your world of I have a very clear through my astrology and through readings I've had and through my own instincts and through just all these different and my own sort of life review making a very practical life review of recognizing that I don't deal with scheduling. I don't I don't follow schedules I create them and love them and then I never follow them. And so like, I've had this. These indications through a lot of my stuff of, I'm not supposed to do it that way I'm not supposed to create a schedule that I, you know, really work into and sit down with and but I'm also it's also not about like sort of this follow your fancy like it's not that it's, I know it's not that because as you said repeatedly that does not work. And so it's, I like that I like I like that phrase I haven't heard that before follow your fancy I don't know where that came from really catchy and I like it. Well, and the thing is, is that I think there's a big misunderstanding between following your intuition. Yeah, and following your like impulses those right different. Yeah, and if I am following my guides through my business there are plenty of days where I'm like, No, I don't want it I don't want to do, really, I don't want to do that and like. And so it's not about like, where do you feel like being today it's about like, it still has its own sort of structure that you have to work with. But it's not the same as trying to create blocks of time and so it's like I've got this one thing over here and then I've got my George, like system over here going oh but you should schedule and I'm just like, Oh no. So what it's really about is right now for me is trying to find how to follow this guided direction and going in 100% because I've been feeling it for a long time, like, like years at this point and when I do it. It all works out. You know, like, I had my business kind of like, slowly diminish. To the point of being almost nothing. A couple months ago, right before I discovered that my mom has massive memory issues and I needed to spend all of my time, moving my mom into my house. And so it's like, that I couldn't have planned that and then I was resisting that happening, but but I watched it, and I'm like something's coming and I felt that something was coming but I, I had hoped it was in my business. And it made a huge amount of space that I would have been a disaster if I had tried to do all of that while holding on to the amount of sort of in and out business that I had before. So it's like, well, I need to lean into trusting it. But that's super hard for me. Yeah, so, yeah, so I need to go in 100% and it's like it's I feel like a, I feel like a traitor in the George world of like, no, no, no, no, but I'm going to follow my intuition instead. Well, and I'll just come in here and say, I love rebels, as you probably know, because I myself am one. I love marketing space and a little bit into productivity. I talk, but, but I love rebels and I think of myself. I did this analogy just came to me, kind of like a, what do you call those? I just came back from vacations on the vacation brain. When you jump off a pool. What's that? What's I think diving boards diving board. Thank you. So I see myself, my teaching is a diving board because it has to be stiff enough where you can actually stand on it, but it has to be flexible enough where people can like jump and do their own thing and do their own bounces and things like that. So I, so in other words, yes, I want you to Sonya I want you to bounce against the diving board of my productivity teaching so that you can do your own flips and land really cool way that's all you can do. But I do feel like, like, we talked about this a little bit before like you're creating some kind of system structure might might we say that other people can probably benefit a lot from. Yeah, if they had the kind of clarity that you do as you develop this thing, because it's still a I mean, no matter what absolutely a system it's a ritual it's all of those things. And I think that it's like this urge that I've had to make it be scheduling blocks on a calendar. It's not about time. And that's, that's the part that is hard for me because I always, I always try to come back around to time, or, or, you know, there was, isn't there a word in Greek for time that's not calendar based but it's more opportunity based is there. Now, those of you watching below, please comment below there is a Greek word I studied this years and years ago. This was actually coming out of, I used to kind of study the Bible more than a lot more than I have now but, but yeah there is a word for like opportunity that's not calendar based. It's more like the, the some kind of yeah it's like, it's like destiny or like the rhythms of life or something like that kind of thing you're talking about that there, there wasn't there wasn't there was a time for your caretaking to start happening that that wasn't like, it's going to happen on September 1 or something like that no it's it's going to happen when it happens but you sensed into it. That was what's really cool about it. And I feel like that's, that's a system that I would love for you to develop and teach to like, because, because your woo is strong. You know, like your sense of that time, those opportunities that intuition that the rhythm that's coming into your life is probably stronger or more aligned more sensitive to it than a lot of people who think they're following their intuition. Yes, exactly. Yes, that's so much that's such a huge thing for me of like, we have what we think is our intuition that is often a blend of our intuition mixed in with shoulds and supposed to use and they kind of clash with each other. And it, and especially if you're also seeking like spiritual guidance from from whatever it is that's happening up there, like, because that can also have its own. Like, it's okay to have your spiritual guidance conflict with your own sense of direction, and you don't have to always take what they say and you also don't always have to listen to your I mean it's messy, it's messy. And so it's so important to start to figure out like, where are my shoulds and where are my have to use and how are they interfering with what I'm actually being given, because let me tell you, the idea of following my woo is so terrifying to me.  I was 100% in that space. And so to have you say like my was strong I'm like well it can't be because I haven't been doing it long enough, or I don't have the history for it or I have too much doubt or all of that, you know, and so I just hit like 20 different topics at once but that's great. What were we talking about. No, no, no, no, it's exactly this. I mean we're talking about, well we're always talking about the authentic business and that's exactly what you're continuing to sense into and build and develop because. Okay, so. And so what I when you talk about like wanting me to like come up with this system or whatever that's really what I'm kind of hoping for out of this year is that I need to dive into it, I need to be all in, and either it will work. I'll be able to be like, hey, I developed the system and also I successfully rebelled, or it won't work. And I'll be like, Oh, okay, that didn't work the way I needed it to and I need to bring in these other pieces but now I know. Now you know from first hand experience. Absolutely, you do need to dive into it fully like I said I love rebels by myself and one, and I would never want someone to go. My guidance is leading me this way but George is saying this thing my guidance leading me this way I would be, I would be horrified if someone says, I'm going to follow George's guidance and not my own guidance. Yeah, because then you're not being, you're not being authentic, the authentic business, not being authentic to yourself.  Yeah, I like being some now some people. I was just gonna say I think it's springboard right you want to springboard but yeah there's also another but like or or or like I guess I guess there was a maybe maybe I'm mixing words here but like in a in a in a drama, like there's there's like the the enemy or there's like the, there's like a person protagonist has to have a, what do you call an antagonist antagonist or has to have someone that makes a protagonist go on the journey and become strong.  Yeah, so so so long story short, I'm really because what when when you do come up with your system, or at least your deep learnings based on your experiences. It's going to be so helpful for all of my job for productivity people as well. So, okay. Well, and I just want to, I want to, I really want to emphasize what you just said to because I think it's so important that people get into these spaces of learning places where they have their leader, and they get really stuck and I see it sometimes in the group where where somebody gets really stuck on like but George said this, but I'm not good at that and it's like, that's so important to allow space for that's one of the reasons why I've heard you and I think it's so important to really bring in to this space of like, this is my teaching and use it to go bounce into your own space whatever that bounce word is. Right. Yeah, which which may go seem to go against my teaching but I am one human being, you know, so had some experiences, I know 0.00 can, there cannot be enough zeros, 1% of reality, you know, and it's like, it's like, like I said, I sometimes sound really certain and persuasive, because I want to challenge people to listen to their own intuition, then go, does that, does that feel right, or does that feel off, and I want to do my own thing. Great.  Let's, let's, let me bring that forefront into. Or it's different, right. It's different. It's for different people and if you're, if you're this kind of person, you should absolutely use Sonya system and, and they'll probably need to borrow parts of it. But, but, okay, so this is, this is really great and I can't believe our time is, is coming to a close. I want to do. Yeah, no, it's great. I love it. I love it. I want to, I want to make sure we cover a couple things one is of course I want to, I want you to tell people about your work a little bit more than just your one line intro earlier, but I also want to hear what your priorities are that we can check in on in our next interview with you. Yeah. My priorities are absolutely to have just really stuck with this ritual this morning, because when it's it's in the morning, it's all about if my if I have constructed the beginning of my day where I check in with my guides and I chat with them and find out what's up. The rest of the day has its own path. And, and that's the thing I need to do that all the time because when I'm just doing it a little bit here and there it's, it's scattered. I've also just recently worked on getting really organized in the last like week, it's been like, get my stuff together get things automated get my processes a little more tech or automated because it was very manual before and I was really annoyed. Because I'm tech enough that I really need it to be like a little more, you know, functional. So I just finished that literally last night where I had everything I have, you know, people can purchase and it's going to automatically go into notion and then notions make tickets for me and I got it so I'm excited to use that. So using that. Um, so it's really about checking in every day and and following having faith. It's about having faith for me consistently or leaning into that that's what I'm trying to work on. And as far as what I offer.  Like, like that that that should quote unquote be the priority for all of us right like we need more faith in our higher selves to guide us and like to become more sensitive to that and to and to follow that guide. I've been super programmed in this society to to be to be really like science based and and I bought in 100% and I still do in a lot of ways which makes it makes it tricky but that's its own conversation. Um, so a lot of what I try to offer is like a grounded we've talked about this having a grounded practical balance between spiritual and I think this is part of the problem is that is that people take off with spiritual stuff into like, what are you talking about, you know, and they just disappear into a new zone that is left behind this existence here on earth. So, having this spirituality that connects us to our experience. And so I work with guides I have tons of guides who I actually talked to and you know that's something that can be practiced and I firmly believe that everybody can do it. Like, when I started, like, I would get like, and I was like, well I heard, okay, great, I'll write that down. Manageable for everybody. Maybe it means something profound and some other. It probably does. I was actually, yeah, I probably was like, hey, you know, your language you need to tone it down a little. So, guide work and tarot and like divination and actual spell work, like working with physical tools we have a lot of language right now that says we should be able to do it all in our minds. And I'm like, no, no, no, no, we are here. We have tools we have physical existence and we have these physical things that help.  So let's also do spirituality with these physical tools that we have around us so I do lots of physical magic. And it's all really fun and it's always about helping people get clear and get either sense of direction or purpose usually those two things. Like you kind of need to get purpose kind of happens first and then your direction is like whoa what happens you know I'm still struggling to understand my purpose, or they already have purpose and they're not trying to focus on that they're like I just don't know how to go in that direction so I'm usually helping people with one or the other. I think there's a lot more about that so give me maybe a little imaginary story of an ideal client who is struggling with something. Which one. Yeah, let's let's let's do one of each how's that. Okay, so if somebody is struggling with direction. Well, I'm going to use myself because it's really easy, which is, which is that I knew I well. Okay, so actually I can be both because I knew there was a point where I felt like I didn't have direction and what I used to describe was.  Yeah, and I was like it's like, I'm this light, I have this light that's going out going where, where's the path, I can't see it. And so I was in my job and I'm just plugging away and it was very dull.  So I started to get these glimpses right I needed some kind of sense of where is the path in my world I don't even know where it is. And then as I started to get into I you know found discovered like coaching as an idea. And as I started working with that I figured out through multiple people who were spiritual helpers to me. I started to learn how much spirituality I wanted to have involved in my work right and that became understanding direction. So, yeah, I'm sorry I understand the purpose. So that it's just kind of back and forth between those two. And I think. Briefly can you just can you because you're using these two terms purpose and direction. It sounds to me like you're making this important distinction between the two can you distinguish that just briefly here. Why am I making a distinction between those two. No, I think I think that I know I feel it but I'm realizing I'm intuiting the distinction, but I just want you to articulate it if it's possible to maybe it's not yet possible. Okay, so I think part of it is that purpose feels like it's your values and it's it's your and sometimes, especially when you're not working in it yet it feels like it's over there out somewhere else and but it's it's it feels sort of like a destination. Yeah, okay. Right. And the direction is like, well, how do I get there because it's not a straight line it's like yeah it's over there that doesn't work you don't get to just jump to it. And so direction is like well right now I need to go this way and then I need to go that way. So in business direction is the path. Right in business language sounds to me like vision versus strategy or something like that. Yeah, the vision of a business is like. Yeah, and those words just make me like. So yes, we were coming up with more more woo friendly language or more authentic language perhaps to oneself so yeah no thank you because a lot of the time people say I feel like I understand this about myself, but I don't know where to go now what do I do with it and that's That's part of what you do is the absolute you love I just I clear away a lot of that and all of a sudden they come out of these sessions are like, Oh, I need to do this I need to do that now. Now just clear like what aspect of strategy or direction can you help with because I because well personally, I know how tech savvy you are. And I know how business and marketing savvy you are as well. No, you are you are savvy on all of these things. I know. So, when you work on direction, what kind of work are you is happening there. I mean it's mostly yeah it's mostly career and business. Yep. Yeah, guidance mostly because I'm not in the health and relationship fields like I just does and that's how it's like well then I guess that leaves me with career direction and yeah. Okay, that's very, that's very good. Well, to be continued. Yeah, looking forward to it. And I hope you okay so if you could summarize the commitment you're making for us to check in on in the next interview. I know putting you completely on the spot but no, no, it's good. How might you how might you summarize what we did what we can check in on. Um, it feels so stupidly simple but it really is like how'd you do with those morning. Yeah, guide check ins. Yep. And how's your faith looking like. Yeah, it's those two things. Awesome. Awesome. And how is that I guess manifesting in your. Yeah, days. And how is it like, how is it showing up in the business because I'm, I'm tired of the so I'm I mean there's always soul searching but like I'm done with the nose dive as far as my or the or the belly belly what do you call it a belly, belly staring or something. Yeah. And so I'm also like, how is it working right is it like functionally practically speaking right how is it showing up in the business and is it having an impact. Cool. Yeah, very good. So folks follow Sonya's journey. And just to be clear, pronounce your name for us just so that I could be validated. You've been doing it this whole time I was really I was like yes he did it right. It's Sonya. Yeah, so follow us. Yeah, follow our journey in the links below. We'll put whatever links you want to have there. And, you know, expect another interview in a couple of months. So, well, thank you so much Sonya for doing this.


Live from San Francisco, it's theCUBE. Covering Google Cloud Next 19, brought to you by Google Cloud and its ecosystem partners.  You're watching theCUBE, the leader in live tech coverage. And we're here at the new improved Moscone Center covering Google Next 2019. I'm Dave Vellante with my co-host Stu Miniman. Stu, good to see you. Russell Warman is here. He's the head of infrastructure at AutoTrader UK. Russell, thanks for coming on theCUBE. You're very welcome. AutoTrader UK, not to be confused with the US AutoTrader. You guys are separate companies. Yep, completely separate. So we operate the largest automotive marketplace in the UK. Something like the 16th busiest website in the UK. Around 55 million cross-platform visits each month. On average, probably about 500,000 vehicles that we're advertising out to our consumers. Yeah, AutoTrader's awesome website. If you're, it's our second biggest purchase, typically behind the home. And so Stu was asking you earlier if digital transformation meant anything to you and you said, well, we kind of went through it 10 years ago. And it's kind of true. You guys are like born in the internet company, but talk about that. So our heritage is we've been around for 40 years. We started producing magazines back in 1977. And then in 96, we launched our first website and we basically just migrated those revenues from our print business online. And then about 2013, we stopped printing magazines altogether, so we're a truly digital business now. And people talk about digital transformation, oftentimes in the context of data. So maybe talk about some of the things that you're doing in your business, some of the challenges that you face, where data fits in, and then we'll really get into it. Okay, so we've got a couple of areas where we've been on premise for our data centers. And we've had that strategy for probably about 15 years. And we've started to reach limits in terms of how we manage capacity within there. And what we've found is using cloud services has really enabled us to unlock capabilities, particularly around things like data. And then more recently, when we're thinking about using Kubernetes and SDO, we've been able to take advantage of things like some of the security features, mutual TLS and service discovery.  You're head of infrastructure. What does that mean to your business today? So we basically run the platform that runs AutoTrader.co.uk. And we're responsible for making sure that's available 24-7-3-6-5. So, and we want to make sure that we're able to give our internal customers the ability to release new applications, new features as quickly as they want, but also that we're ensuring that our consumers, our retailers, get a great experience that it's fast-performant and secure. Yeah, and give us, tease out a little bit, just kind of the scope, how many people you have and how do you balance the stuff that kind of is under your purview and the stuff that you manage that is outside of your four walls? So there's about 25 people within the team. There's probably about 10 people building the infrastructure and about 15 people that are responsible for monitoring it. So we're not a big organization in terms of operations infrastructure. And we have to work very closely with our product squads, because we manage some elements of it, but we're dependent on them managing other elements as well. So we're trying to unlock platform capabilities so that we've got consistency that we're trying to provide those capabilities rather than going off and finding something else. You've got infrastructure on-prem and you've got infrastructure in the cloud, just the cloud services, right? Yeah. So oftentimes those aren't the same. No. Right, talk about your situation and you're on a journey, I presume. What's that journey like? So we started about 14 months ago building out our cloud services and probably in the last four months we've really started accelerating that migration plan of applications moving up into Google Cloud. And our ambition is to get that done in 18 months because the complexity of managing multiple environments is something that we want to try and avoid and minimize that. So is the aspiration to go all in? Absolutely. Absolutely. We've taken a view that for services that we can, we'll take managed services and then everything else will be cloud first. All right, I believe I heard you say Kubernetes is in the mix now. Can you explain where that fits, how you're using it? So that's the platform that we're using to build out all our applications in the cloud. So that's the sort of landing point. Okay, are you using any Kubernetes in your data center today? No, just all from a cloud. That's the platform layer that you use in the public cloud. Yeah, absolutely. So when we talked about should we use Kubernetes we did discuss should we do it on-prem or do it in the cloud? And we just figured that we didn't want the overhead of managing it ourselves on-prem. We thought it'd be better just to take it as a service almost and manage it up in the cloud. Most of the shows we go to, many of them anyway, Stu and I and theCUBE, they talk about multi-cloud and we often say multi-cloud is a function of multi-vendor. Certainly we heard today from Google about multi-cloud. What's your cloud strategy? Because there's another camp that says, well, if you do multi-cloud it's more complex, it's less secure, it's more expensive. Are you trying to be kind of unit cloud or is it horses for courses? So we're with three clouds today. So our data platform sits partly in GCP but then partly in another cloud provider. And then we're also for our enterprise applications we're using another cloud provider because it makes sense to do so. So we want to use the right cloud for the right applications. I think that most of our customer-facing applications will end up in GCP but some of the back-end services may end up in other cloud providers. Okay, so it's strategic fit based on the application. Absolutely. And so what is the wheelhouse of Google in your view from a customer perspective? So the fact that Kubernetes was developed by Google is a really strong play there. It gives you confidence that they know how to look after it. Things like STO have made a massive difference to our organization as well. Part of the reason why we've ended up using GCP was based on using STO and that was around solving a specific problem that we had. So those are the sort of two sort of areas that we focused on. Go ahead. I'm sorry, just when you look at, you've got that hybrid multi-cloud environment that we find is pretty much the standard today. Can you speak a little bit of the management layer, how you look at that? Is it good enough today? And what could the industry do to make it even better? It's good enough for our needs. I think that the challenge that you've always got is data ingress and egress between clouds as you want to move data or query data and making sure that you can do that in a secure way. That's probably the biggest challenge that you'd have and also around the cost of doing so. You know, those are probably the bigger challenges. And the challenge is the cost and time of moving data? Is it security or the above? Maybe you could dig into that a little bit. It's probably a little bit of all the above, if I'm honest. I think you can do so much to secure it, like private VPNs between your cloud providers. It's, I guess, it's the time and actually moving some of that data between the clouds that's a challenge. And then the cost, like I say, it's difficult to predict how much it might cost you to move some of that. What are the big challenges from a business perspective that are driving your technology strategy? Obviously, you want consumers coming to your site. You want to make that as rich as possible for them. You've got a monetization strategy as well, but talk a little bit about the business drivers that are affecting your technology. Okay, so consumer demand is constantly changing. Technology is massively disrupting how people think about and search for cars. We can see that there's a demand in the marketplace for people just to be able to choose a car and have that customized and delivered. They want to have complexity around how they look for finance on their cars. They want all that sort of taken away. They don't want to have to turn up to a car showroom and then go through the same conversation that they've with a salesperson that they've just gone through online. They want that journey to be seamless. So there's some challenges that the industry faced in trying to do that. And where we're trying to help our retailers is by providing those services that customers want. So to be able to understand what the par-tex valuation is on their car before they turn up to the dealership and know that it's going to get honored because they've got a trust in it. So they can understand what they can spend on a monthly fee and what cars are available. So we're trying to change what we provide to our consumers and also to support our retailers manage that change as well. Yeah, Russell, I think it's a really great point you make. We know many industries and especially your industry is changing really fast. So what do those stress and strain mean to the infrastructure team? Can maybe talk a little bit about the relationship of kind of business to IT and how that's changed in the last few years. So probably over the last 10 years, IT's always been seen as the sort of the blocker in terms of making sure that new products and features can become available quickly. And our role is really to make sure that we're providing the infrastructure at the right time for people so that they can basically just dial it in when they need it. So if they want to release a new application, they don't need to come and speak to somebody in ops to sort of provision a server. They just create a pipeline, deploy their application and then the server's provisioned at the same time. And we do things like checking for OWASP vulnerabilities at the same time and making sure that the application and the infrastructure is working hand in hand. So we've taken a lot of that dependency away from people sort of with handoff points and everything.  I think you said you started in the 60s. 70s as a publisher, basically paper, printing presses and the like. 1996 is when you went online, is that right? So that early days pretty much of the internet sort of the heart. And so you've proven that you could transform forward thinking. I'm wondering what the conversation might be like inside the company about things like autonomous vehicles. Is there a day that comes where owning your own vehicle is the exception rather than the norm? What implications that might have for your business? It's very hard to predict obviously, but are you having those kind of conversations? You have other lines of business that you're launching to? So you're absolutely right. Those things are an area of concern and we've got people constantly looking at new and emerging changes and threats across the whole industry. And they are absolutely looking at how that might impact our business. What we're actually seeing though through research that we've done is people still aspire to own cars. So car ownership is still something that people want to look at. With regards to the second part of your question, we are automotive focused. That's where we are. We don't operate outside of the UK and Ireland. We absolutely focus on our core and making sure that the services that we offer around that really support our customers' needs, whether that be consumer or retailer. Well, it's interesting. The auto industry really hasn't been wildly disrupted. I mean, you certainly see Tesla and people talk about autonomous vehicles, but still the big car makers are still doing quite well. They dominate the market. So it's going to take some time and there's some skeptics out there, but it's going to be interesting to see how you guys evolve. What other opportunities you go after? I want to ask you, so Google Next is coming to London in November. How important is it for you as somebody who's based in the UK to have Google have local events like that?  How important is that to you as a customer? It's important for us to be able to get more people involved in the conversation. There's obviously a cost and time effort in terms of coming out to something here, but it's, I mean, the scale of this is enormous in comparison to what we'd see in the UK, but it just means that having that presence in the UK means that we can just get more of our engineers in front of the right people to have a better conversation, understand more about what's coming and how we might be able to use that within our business. Things that, so you've just sort of, you're early on in your cloud journey, but knowing what you know to get, it's early on. Things you might do differently if you had a mulligan you want to do over again, or advice you'd give to your colleagues and peers? So I think the big thing that we found is modernizing applications before you start moving them into the cloud. I think there is a tendency probably, and we were in the cloud probably about nine years, 10 years ago, and we bought those workloads back on-prem, and the reason why we did that is because we didn't treat them differently. We managed those instances in the same way that we managed on-prem, which is completely the wrong way to do it, in my opinion. So we needed to change our mindset in terms of how we managed the infrastructure.  We talk about there's a lot to do of changing the operating model, right? The long pole in the tent is, modernizing those applications, any change, migrations is really tough, so do you have a timeframe? How long will it take you to get to that all-in that you're planning on? 18 months is our target. That's our target. And not just a lift and shift. No, no, no, so that is a modernization. But the approach that we've taken and is to try and give the people the confidence and the courage really to go at it is we've picked off some easier applications, so this journey about being cloud native, we started six or seven years ago and we've started getting people building their applications on our private cloud and those ones have been really easy to move on to the public cloud. And some of the applications that have been a little bit trickier that have been on our sort of traditional virtualization platform, we're trying to pick those off in parallel, so you do one that's really easier, one that's a bit harder, so you give people confidence and then you're trying to solve some of the tricky problems alongside it and I think that just incrementally gives you more confidence that you're moving in the right direction. It's a great strategy. It's like skiing on the blacks and then going to the greens and then making your moves. Just a little bit harder each time you go at it, but yeah. Russell, thanks so much for coming to theCUBE. It was great to have you. It's a pleasure, thank you very much. Okay, thank you for watching. We'll be right back from Google Next in San Francisco.


How much did you sleep when Wrath Fresh? I mean, nothing until I hit max level. That was longer than this, so this is like half. Not even that, probably. Can't feel my hand. Stop inviting me. Did anyone manage to actually make a character on Living Flame? Or is everyone still just trying to log in? I got two logging into Game Server. I was quick as well, man. Come on. Oh, dude! Wait, wait, wait. Positioning Q, though. That's not too bad, right? Positioning Q.  Is it gonna be like, yeah, Suzzard, Suzzard, mates. No characters for you. I told you, man. Server first, coming up. Hairstyle bold. Looks good taken. Motherfuckers, okay, two A's. Okay, we're good, we're good, we're good. That's good. Can trade 40 to 60 water for one. What are you, water?  I don't need water, man. I need linen. You got linen. We got a deal, but you need to bring me course thread as well because I can't make any more robes without it. And I also need some simple wood.  So it's like... Equal total. Two priests, 15 to 16. Okay, nice. We're catching. Now we do a pumper level. What's this shit?  I find Mancrix's wife. Sorry. What did he say? Uh, desecrated citadel is a monkelo. Hey there, just passing through. Let's try healing focus, see if it helps with the penance pushback. It did. We need to make this one run. Ha ha. I need to rest someone, bro. It was all planned out, man. It was all planned out. All right, this quest here should be me, man. Easily, actually.


What is responsible innovation and why is it important? Of course, no one sets out to innovate irresponsibly. Most innovators want to solve problems to improve lives and to make the world a better place. But all too often, good intentions end up causing harm as the best laid plans go awry. This may be due to sheer bad luck or an excessive tunnel-visioned enthusiasm, or it could be the result of overlooking the collateral damage that all too easily accompanies half-baked ideas.  This is bad news if you're someone who's at the sharp end of a good idea that ends up benefiting others at your expense. But it's also bad for business if, despite your best intentions, you end up being branded as irresponsible. So how do you ensure that good intentions in innovation lead to new ideas and capabilities that achieve what you aspire to without causing unnecessary harm? One place to start is with these five risk-bite rules of thumb. 1. Remember that the first rule of responsible innovation is to talk about responsible innovation. It's amazing what a difference it can make if you take the time to talk to others about how best to be responsible. 2. Think through the possible social and ethical consequences of what you're planning and how you're going to navigate these. 3. Consider what's important to you and the people your innovation will touch, and importantly, think about how you can stay true to your values while respecting theirs. 4. Think about how your innovation might exclude or discriminate against some people and how you can work with them to ensure that this doesn't happen. And 5. Plan ahead for how you'll handle the inevitable yet often unexpected downsides of your innovation. And think about how you'll change direction when this happens to ensure you end up doing more good than harm. The bottom line here is that innovation always has its downsides as well as its benefits, and with the best intentions in the world, the chances are that at some point you'll make a mess of things. But by being proactive about innovating responsibly, you'll increase the likelihood of success not just for yourself, but for the people whose lives are affected by what you do. For more information on responsible innovation, please do check out the links below. And as always, stay safe.