So having one box that we've made, or one model building that we've made is great, but I chose a certain size that I wanted these squares to be. So I'm wondering what's going to happen if I choose a different sized square. Let's see. Maybe to start out my experiment, I'll make a rectangle that has little tiny, tiny, tiny squares cut out of the corners. [BLANK<u>AUDIO].</u> So now that I have my tiny squares cut out, I'm going to cut out some really big ones. Actually, I'm going to consider, first of all, this box as our medium-sized corner box. And for comparison, I'm just going to take that tape off of it. [BLANK<u>AUDIO]</u> . So medium one is back. Now for the really big one, or rather the really big corners. [BLANK<u>AUDIO].</u> So now, we have three very different looking rectangles with the corners cut out of them. And the question I have for you and something that Athena's wondering is which one of these has the biggest volume when it's folded into a box? Or in other words, which one's going to hold the most M&M's? Clearly we want the box that's going to hold the most M & M's. So pick between these three boxes.
So it turns out that this middle sheet of paper is going to make the box with the biggest volume. But in order to really visualize that you have to put the boxes together. So let's start out by doing that. Okay, so now we have three boxes and I think it's time to make use of our M&Ms now. [NOISE][SOUND] Okay, so I've filled each of these boxes with M&Ms as best I can. Unfortunately, I ran out over here, probably because I ate some of them. But I think we can already tell, just by looking at the dimensions of this, little skinny box. That its volume is definitely less than the other two boxes. It's so flat, it can't possibly be the biggest. So now that we're left with two choices. Now I could go through and count every single M&M in this box and every single M&M in this box. But, I'm kind of lazy and that would take a really long time. So, I will tell you that when I filled these M&Ms it took me several more handfuls of M&Ms to fill this box than to fill this box. So that means that its pretty good guess that this one has more volume. However as with many things you see in this course I think we're going to need to do some math to find a more rigorous way of analyzing the volume of any kind of box that we could make out of a piece of paper.
Volume. So, what really is volume? It's a word we hear all the time, but in math it has a special meaning. Just as we think about it in everyday life, volume is a measure of space, 3D space to be precise. So let's say I have a balloon, like this one right here. It's not inflated right now, it's pretty flat, not actually just two dimensional, but there's no air inside of it right now. To puff it up. So right now it's not really taking up very much space. However when you inflate a balloon, you increase its volume because it takes up more space. The volume of a balloon is hard to calculate mathematically. The volume of a box Is pretty easy though. Let's take this box, for instance. We can say that there are 3 different lengths that we can talk about with this box. If I tilt it a little bit, you can see its height, this length right here. You can also see the width, which I'm going to call this dimension, and we can look at the length too, right here. Since this is kind of bulky to work with right now. I'm wanting to draw my own rendition on the tablet. So here is my very fancy drawing of a box. You can see I've labeled the length, the width, and the height of the box, the three different measurements that we can take of it with a ruler. Now for a box, the equation to find the volume is quite simple. The volume is just equal to the length times the width times the height. In fact, I'm going to go ahead and label these sides with these abbreviations. So here we go, one side length of l, one of w, and one of h.
So Athena's been building a lot of models in her architecture class so that she can depict buildings. And she figured out one really easy way to build a rectangular building. All you do is you take a rectangular sheet of paper, and you just cut a square out of each corner. So you cut a square out of all four corners, and make each square the same size. So let's try it out and see what happens. [SOUND] Voila, four squares drawn Okay, great, so now we have a rectangle without any corners. Now all I have to do is fold up the different flaps. And look, we have a box. If you want to make it a permenent box, we can always tape each corner together. And look at that, we have a building. Now, this may look pretty simple right now, but If you think about it, we've only dealt with area in this class so far, and that's really all that Athena's talked about in her architecture classes so far too. But being able to make a box or a building, or whatever you want to call this, is really powerful. This is a three dimensional object. Means it has space that it occupies, and I can do stuff with it like[NOISE] Fill it with M&Ms. [SOUND] So this could be a model building or it can hold stuff. It can hold M&Ms, it can hold my tape,[SOUND] I can do whatever I want with it because this shape takes up volume. It doesn't just have area, it has a volume. So this is a totally new concept for art class and also a new concept for Athena. It's going to be super important as she continues to study architecture.
So, let's say we have a box the height of 5 centimetres, a length of 10 centimeters and a width of 8 centimetres. Then what is its volume meassured in cubic metres?
Well we know that the formula for calculating the volume of a rectangular prism or a box like this, is just to multiply the length, the width, and height together. So all we need to do is plug in the values that we know for each of those quantities. So we'll have 10 centimeters times 8 centimeters times 5 centimeters, and that gives us a final answer of 400 centimeters cubed. Or, in other words, 400 cubic centimeters.
So what did we learn from the last quiz? Well we've seen that volume of our box changes depending on how big the square that we cut out of each corner was. So that means that volume in this case is a function of square size. That's pretty cool. Now for Athena, we're going to have a little project. I want us to figure out how to maximize the volume of a box that we can make out of a sheet of paper this size. The same size we've been using which is 5 inches long by 7 inches wide. But in order to do that we're going to have to start by learning a little bit more about volume.
Okay, so we know how to calculate the volume of a box if we know the length, the width and the height of the box. So let's take another look at how Athena was folding her boxes. She'd take a piece of paper like this one. Then she would measure a square in each corner of her rectangular piece of paper. Then she cut the squares out of the corners. And then she would fold up the edges. And tada, with a little bit tape we'd have a pretty nicely put together box. So we know then we can make box out of a piece of paper that looks like this. It once was a rectangle until I cut the squares out of the corners. Let's draw that on here. So here is a nice little piece of paper for us to start out with. And then I can draw this to show what it would look like after the corners have been cut out of it. Awesome. Now, on the real piece of paper that I was using, this one, the original side length of this long side was seven inches. And the original side length of the shorter side was five inches. So let's pretend like this paper has the same dimensions. So the orginal side length was five inches. And this original side length down here was seven inches. Now, Athena and I are both super careful people, so we made sure that the squares that I cut out of each corner are exactly the same size. That means that each of these lengths of a cutout square is the same. Let's call that length x. Now, I'm going to give you a second to refresh your memory on how this box looks when we fold up the sides. So here's my real one again. This length is x. This is seven inches and this is five inches. When I fold it up, it looks like this. We have length, width, and height. Now, my question for you is, once I fold up this box, what will the height of the box be? Please put your answer in this box. And just a hint, it may not just be a number.
When we fold up my real box over here the length that becomes the height is this shortest length, which is the side length of each of the squares that I cut out. That's just x over here. So the height of the box is going to be x inches.
So, since this is going to be the height of the box, h or the height, is just equal to x. Remember that we were calling this longest side the length of the box. Can you tell me then, in terms of this initial length of seven inches, and in terms of this cut size x, what will the length of the box be once it's folded up?
The length of the box, since it's folded up, would be 7 minus 2x inches. Since the initial length is 7 inches, and then from each corner, we take out x, we're taking 7 minus x minus x, which is just equal to 7 minus 2x. So, that means our new l for the box is just 7 minus 2x.
And lastly, we can figure out w, the width of the box, the last dimension that we want to look for. Originally, the width was 5 inches. So, what's the new width going to be once the box is folded up?
We started out with five inches. And just like we did with the length, we just subtract x from each side. So, we're going to have 5 minus 2x inches. Then we have w equals 5 minus 2x. Cool, we have all of our dimensions covered.
Now that we have all three of our dimensions in terms of x in these original constants, we're ready to calculate the volume of this box. So what will the volume of Athena's box be as a function of x. Please fill in that expression over here.
All we need to do here is starting with this equation for the general volume of a box, substitute in each of these expressions for the height, length and width. So we end up with v of x equals the quantity 7 minus 2x, times the quantity 5 minus 2x times x. Now of course you could multiply this out as well. And if you go through the whole task of multiplying everything out, we end up with 4x cubed minus 24x squared plus 35x. Either way though, is a totally fine form. In fact, I almost prefer this factored version.
So, Athena has this equation now to describe the volume of one of her boxes. Let's make sure that know what this function does. Can you fill in the blanks in this sentence? This function takes blank as input and gives bank as output. In either box, I'd like you to put one of the letters a, b, c, or d corresponding to which of those phrases you thing belongs in either spot here.
Option a goes in the first blank and option b goes in the second blank. That means that this sentence reads, this function takes the length of one cut, in other words x here, as input, and gives the volume of the box as the output. We can actually see that from the way that our equation is written. We have x here, as the input, inside the parenthesis, next to the function name. And, the name of the function is V. And the function itself represents the volume of the box. So, that is the output.
So, for example, what would the volume of the box be if these cut lengths, the distance x in our diagram, were 1 inch long? Well, since we know that the cut size, x, is the input for our function, we will just calculate v of 1. And plug in 1 in the spot of x. After some simplification, we would see that the volume of the box, measured in inches cubed, would be 15. That's pretty interesting, but one piece of information isn't enough to figure out what the best cut size is. And remember, Athena wants to maximize the volume of her box. So let's try a few different values. So, I've already filled in the volume of the box that would result if the cut size was 1 inch. And I'd like you to fill in the other four volumes that are left over for the other 4 cut sizes that are left over.
If we plug in 0 as our input we get 7 times 5 times 0 which is just 0. So if the cut side is 0, the volume of the box will be 0. This makes sense because there will be no flaps to fold up since there would have been nothing cut out of the original rectangle. You won't be able to make a box at all and it definitely wouldn't be able to hold anything. So it makes sense that the volume of that would be 0. Plugging in 2 to this equation gives us v of 2 equals 6. Plugging 3 in gives us for the volume of negative 3. And last but not least, if we plug in
But, uh-oh, it looks like we have a problem. We did fill in our table completely, but one of these answers for the volume of a box doesn't make sense. Can you tell me which answer is the one that has an issue? Keep in mind what volume is in the real world, and then how the quantities that it's allowed to equal will impact your decision over here.
This one. The one where v of x equals negative 3. Or rather v of 3 equals negative 3. How can we have a negative volume? That doesn't make any sense whatsoever.
The reason that this volume doesn't work is because it's giving us a negative value. For a quantity that we should be able to touch and feel, it's tangible. And length and areas and volumes are usually positive. Let's see if we can figure out why by using a real piece of paper again. After all, this is the right size, so maybe if we try to make this box, we'll figure out what's going wrong. Here's my ruler and my piece of paper and I'm going to measure in three inches from each side and then in the other direction as well. Okay, so there's one of the squares we should draw. Now let's draw the other three. Two squares, and now potentially for the tricky ones. So here's the issue we have when I tried to draw my boxes for the corners that I want to cut out. This length right here is 3 inches, or x, the length of the square that I want to cut out. But you can see that if I started to cut here, and actually cut down the whole length, right there, I'm already cutting into an area that's supposed to be part of another square that I'm going to, to cut out. This is a major issue. I'm not going to have enough flaps left, or enough material left if I cut out all of this. In fact, let's just do it to see what happens. Now I just have this strip of paper and I can't make a box out of this.There are no flaps. I guess this has something to do with what's going on. So for our 5 inch by 7 inch piece of paper, what value must x, the cut length, not exceed in order for us to still be able to have flaps that we can fold up and make this into a box? Just think back on what the problem was before with the 3 in cut length, and use your common sense. This is kind of a riddle. So, just think through and take your time.
X must not exceed 2.5 inches if we want to avoid running into the same problem that we ran into just a second ago. Here's why. Of the two sides of our original piece of paper, the 5 inch long one is the shorter one. And we'll end up without a flap to fold up from the 5 inch side. If the side length of the square that we cut out is at least as long as half the length of this side, which is 2.5 inches.
Now that we know the upper limit of what x can equal, that brings me to our next question of what the lower limit of values x can equal is. Rather, if we want to make sure that we can still fold this piece of paper into a box, what value does x need to be greatest than at all times?
We need to make sure that x is greater than 0 inches. Now of course if x was equal to 0, then we just wouldn't have cut into the box at all. You would have just a regular rectangle of paper. And this is physically possible, however there are no flaps here to fold so it's not really our ideal situation. A negative value, however, doesn't make any sense. After all, like we've seen many times in different examples throughout this course, measuring negative lengths is a pretty tricky thing to do. It doesn't make sense in this situation. So, bottom line, if we want flaps, x must be greater than 0.
Remember that the reason Athena is creating all of these box models in the first place, is for her architecture internship. So, that means that when she's discussing functions like this one, she's probably going to want to focus on the values of the domain that make sense in the physical situation she's discussing. We figured out before, that if Athena wants to make sure, that she's actually able to fold her piece of paper into a box, x should be greater than zero. But it should also be less than 2.5 inches. However, we saw that if x equals zero, then the volume equals zero. And if x equals 2.5, the volume also equals 2.5. The idea of zero volume is actually pretty understandable. It's negative volume that's the issue. So, I'm going to go ahead and decide to include these endpoints in our domain. So, here we go. This is the domain of the function that we've decided upon, using our common sense to deal with the situation at hand. So, like we talked about in the last lesson, this is not a mathematically-imposed domain restriction. This is one that we're imposing upon the equation ourselves to make the equation better fit the situation of the box. So, let's visualize this with a graph using this domain restriction. So, here's what a graph of this function looks like with these domain restriction put in place. We can see that the graph is only allowed to take on values of x between points, so they are closed circles to either one.
Thinking back to the beginning of the lesson, remember that Athena was interested in figuring out what value of cut length or x, would maximize the volume of her box. So the question I have for you is, on this interval that we've decided are possible values for x in our situation, which of those values leads to that maximum volume? See if you can figure out a ballpark range on the graph, of where you think that maximum point is. Estimate what the x coordinate of that point is, and then pick the point over here that you think is closest.
The answer is 0.959. Now on our graph if we're looking for the maximum point, we travel up, and it looks like the peek is right about here, just to the left of the line x equals 1. That means that .959 is the closest answer. So what does this tell us? Well, this tells us that if Athena cuts squares of side length .959 inches out of each corner of a piece of paper, the box you would get out of that would have the highest volume that any box you could make out of that initial piece of paper could.
Now that we've figured out the horizontal position of the maximum value of the function, in this interval at least, I'd like you to find that maximum value of the function that we're looking for. So, find the y coordinate or the f of x coordinate of this peak point. Is it about 0, 2.5, 10.243, 15.02, 25.6 or 13.67.
There are a couple of different ways that we could do this problem. For one thing, we could go back to the graph like we did the first time to figure out about where we thought this peak was, but since we already have an x value for that peak of 0.959, we could also plug this value into the equation for V. We could use it as a value for the function. The general way we do it graphically or by hand, we can see that the y value of this point is about equal to 15. So,
In this problem we wanted to find the maximum volume, and we successfully did that. We found the coordinates of this top point for this portion of the curve. Finding maximum and minimum is something we're often interested in, in math. It's a matter of identifying the point on a graph that is highest or lowest on the vertical axis. In a given area of the domain. So what is the minimum of this function? Please fill in the two blanks right here. Tell me first what the minimum value of the function is, where we can tell the value of the function based on the vertical axis here, which represents f of x, and tell me then what the x coordinate of this maximum is.
Well right here, is the point that's the minimum, the vertex of the parabola. It looks like its height, or its vertical coordinate is 3 and its horizontal coordinate is 2. Since the vertex of an upward facing parabola represents the minimum point on the entire curve, 3 is the lowest y value, or lowest value of f of x, that any point on this fuction will have.
So, here, once again, is our graph of the volume of a box versus the length of the cut that we put in it, to be able to fold the sides up. And clearly, the one that we have here is the restricted domain version, the one that Athena wanted, because of the real life situation that she was using this in conjunction with. However, let's take a second to look at the equation for this function. We had A of x so you go to 7 minus 2X times 5 minus 2X times x. And you'll remember earlier on, I multiplied all of these out. And when we do this all of this multiplication out, we get a different form, for expressing this function, 4x cubed minus 24x squared plus 35x. And the reason I wanted multiplication out right now, so I think you might notice, this equation is pretty different from anything else that we've grasped so far in the course. In particular, we have an exponent of 3 in our function. What on earth does a graph that has a third degree term in it look like? Well, why don't we get rid of these domain restrictions briefly and zoom out, and look at the entire graph that this is part of. So, here's the full version of our function without any domain restrictions. Multiply it out. The function, itself, is 4x cubed minus 24x squared plus 35x. This is a totally different equation than anything we've dealt with before, graphically, at least, though we did see expressions that look like this early on in the course. Because this function has a different degree than other functions we've dealt with, it's going to have different graphical properties as well. Now, I've labeled a few different special points on the graph or points that may potentially be special. And I'd like you to help identify what they are. So, in each of the purple boxes over here, to complete the statements, I'd like you to fill in letters that go with these different points. And there's this blue one right here, this, I'd like to be a number that you just type in. So basically, I'm asking you which points are the x-intercept and how many are there? Where the function changes from increasing to decreasing, and when it changes from decreasing to increasing.
There are three x intercepts. Which we can see are the points a, c, and e. Now, remember that a function is said to be increasing if its slope is positive. Because that means that the y values, or the values of the function, the output, are continuing to increase in value. And the graph goes from having a positive slope to having a negative slope. At the point b. That's the point of transition from the function increasing, to the function decreasing. And the function continues to decrease, until point d. At which point it switches from decreasing to increasing. Awesome job.
Now we just said that points B and D, are special because they represent a switch in the direction of the slope of the function. Now speaking of general behavior of the function, I'd like you to tell me, what its domain is and what its range is. Now please note that even though I've written y equals this time instead of F of x equals. This is to remind you that the reason an equation is a function is not because of what we choose to call the dependent variable. It's because of the way that this expression over here involving x's maps the inputs to the output. This is in fact a function, and we can write that in a more technical way by saying y equals f of x. In other words the real function of x is this expression over here. The one that's taking all the inputs of x and modifying them to create the output which is y. Just on the side to make sure you understand the notation here, what purpose it serves and how all these different things are connected. So back to the function itself, please write the domain and the range using interval notation in these two boxes.
The domain of this function is all the real numbers, which we write in interval notation as spanning from negative infinity to positive infinity. Now, there are no domain restrictions on this function imposed by the math itself because every value of x that we could plug in from all the real numbers, is going to cause the function to spit out a real value of y as well. So, what is its range? Another good question. Well, we can see from the graph, that as x decreases, y is continuing to decrease as well. And as x increases, y is continuing to increase. If we look at the function, the higher the value of x that we plug in, the higher and higher and higher, y is going to go. And similarly, the lower the x value we input, the smaller this number over here is going to get for the output. So, the range is also all the real numbers.
So, we just showed that the domain and the range function are both all the real numbers. But still, from looking at this graph, points B and D seem to stand out. As we saw before, B and D both represent changes in the direction of, of the slope of the graph. They also kind of look like a peak and a valley. Now, let's compare this to another kind of peak or valley that we've seen before. Now, let's say that in addition to this curve, we have a second one on the set of axes. It's another function, and this time, it's just a parabola, a shape that we've seen many times before. Now, I'm going to label the vertex on this parabola, which we know is a special point for it, and I'm going to call that point G. I've given you four choices to choose from to figure out what is special about points B, D, and G. Is one of them a global minimum, or a global maximum, or a relative minimum, or a relative maximum? Please pick the proper letter to go in each box.
Point B over here is a relative maximum, point D is a relative minimum, and point G is a global minimum. So, just think about what the words global and relative mean here. Relative maximum, for example, when applied to a certain point on the curve implies that the y-coordinate or the output of that function is higher than the values of the y-coordinates surrounding it. The opposite is true for relative minimum. Although the graphicals all the way down to negative infinity down here, there are other points where the graph gets to make the awesome transition from having a negative slope to making a positive slope.
Let's talk about one of the new terms that we just saw in the last quiz, a relative minimum. Now if we have some function, f of x, we can call a value of that function, at a certain point, a relative minimum, if it is related to other points on the function how. Does it need to be lower than them or higher than them? And does being a relative minimum refer to the value of the function at that point, in relation to the entire range of the function, or just in relation to other values of the function near the relative minimum?
So relative minimum is a value of the function that is lower than other values of the function, but just values that are near that point. So, this is why the point "D" on this function over here is a relative minimum since we know that the range of this graph goes all the way down to negative infinity D Is definitely not the lowest point on the graph. In fact we can see, looking at points over here. To the left hand side of the graph. That there are values visible to us right now. That are lower than the value of the function at D. However, compared to all the values adjacent to D. Whether you move to the left or to the right, the function has its lowest value at this spot. This is different from a global maximum like at G. There's no other point on this parabola where the function has a lower value than it does at G.
Let's switch things up and talk instead about relative maximum for a second. So I'd like you to answer the same question as you did before, but this time, talking about relative maximum instead of relative minimums.
Now, a relative maximum is a value of the function that is higher than all the other values of the function that are in the neighborhood of that point. So, a so-called relative maximum like a relative minimum because it's a maximum just in relation to the points surrounding that point. So, as we said before, B on this purple curve is a relative maximum, that's because all of the points to the left of it and all the points to the right of it, in the general area of the curve close to B, have values of the function that are lower than what we find at B itself. So, if we relate B to parts of the graph that are further away, like points up here, we will find some areas where the function takes a higher values than it does with B. But those aren't close by. For the points that B really cares about that are close to it, it's right on top.
So, let's spend a bit of time now looking at this new graph that Athena is using for her box-making project. Now, I think that it's pretty incredible if we take a second to notice just how different this new purple curve looks from this other shape, a parabola, which we're super familiar with by now. Now, if we look at their equations, you can notice that the defining difference between the two equations is that this one, for the purple curve is of degree 3. And this one for the parabola is, of course, of degree 2. So, let's explore what makes this graph different, just a little bit deeper. So, the first things first, are both of these graphs functions? For each equation, pick whether or not it is a function.
And the answer is yes, both of these are functions. They both pass the vertical line test, which we know means neither one of them as more than one output, or y value, for a given x value. That means that we could also represent these with function notation like f of x instead of y. Y is still perfectly acceptable thought.
What do you think the best name, for each of these kinds of functions is? I've given you a list of different possible names here. And for each one, I'd like you to pick one from the list. Your choices are constant function, linear function, quadratic function, cubic function, square root function, or absolute value function.
This first one y equals 3x squared plus 12x plus 20 is a quadratic function. And the second, y equals 4x cubed minus 24x squared plus 35x is a cubic function. Now both of these names might remind you of some stuff that you learned during our polynomial section at the beginning of the course and that makes perfect sense. The names of these types of functions refer to the degree of either one of them. So cubic refers to the fact that this is of degree 3, and quadratic refers to the fact that this is of degree 2.
Now since cubic functions like this one right here, are completely new to us, I think it would be good to get back to the basics a bit. Think about the defining feature of a cubic function. What is it that makes this cubic instead of quadratic? Then write the simplest cubic function you can think of in this box.
F of x equals x cubed is the simplest cubic function that I can think of. Since in order to be a cubic polynomial, an expression must have at least one term that's of degree 3, and that must be the term with the highest degree. This certainly does seem like the simplest cubic function we can come up with. It's just x multiplied by itself three times. Now, because this function is so basic, so primary, this gets a special name, a special designation, of being called the parent function for cubics. Just like f of x squared is the parent function for quadratics.
We just talked about the parent function for a cubic functions, f of x equals x-cubed. So I thought I should show you the graph of this function. Here it is, this blue, pretty curve. You can see that this has the same overall behavior, as the other cubic function that we saw in the last example. That one looked a little more like this. You can see that the overall behavior is the same. One end of the graph is going down to negative infinity, and the other end of the graph is going up to positive infinity. In the middle, there's sort of a grey area of, in this case, going down and then going up again, and in this case, sort of leveling out for a bit. Now as I said in the last quiz, this parent function looks really different from the parent function for quadratic functions. Let's just add that on to our graph, so we can compare them more easily. So here, once again, is the parent function for a quadratic function, f of x equals x squared. Let's examine the overall behavior of this graph as well. We know that the general shape of a parabola is kind of like a u, or if it's upside down, like an n. Whether it's opening upward or opening downward, the parabola has a vertex, which is either its minimum or its maximum. And then both of its ends point in the same direction. And then as x gets further away in either the negative direction,or the positive direction, the graph points the same way. Either both ends of it go to positive infinity, or both ends go to negative infinity. We also talked earlier about how a parabola has an axis of symmetry running down the middle of it. So that if you fold the graph in half along that line, it will exactly map to the other side of itself. That's not the case over here with x cubed. If we simply fold this graph in half, down the y-axis, this right side is not going to end up looking just like the left side. It'll look more like this, not the same. This is all really interesting stuff. Now since we're increasing powers, why don't we just do that one more time. So I've put both y equals x squared and y equals x cubed on the same coordinate plane, then I draw another graph over here for you. My question for you now is what function does this graph represent? Now think about what this graph does on either side of the y axis. And also think about what points you'd expect each of these functions to go through. Your choices are y equals x, y equals x squared, y equals x cubed, y equals x to the fourth and y equals x to the fifth. So you can always make a t chart and plug in some points and see which of those t charts best matches this graph.
This is the graph of y equals x to the 4th. Now because this power is even just like x squared, there's no real number that we can plug in for x, that's going to give us a negative answer for y. For example, negative 3 taken to the 4th power is just four negative 3s multiplied together. And because there are an even number of negative signs here, they're all going to cancel one another out to give us a positive answer. Sure enough, this is positive 81. So what this parent graph, y equals x to the fourth, of fourth order polynomial functions, the lowest y value that we can have is going to be this point right here, when x equals 0. This point is just the origin: zero, zero.
So, we still have x squared and x cubed functions graphed over here on this coordinate plane. And here, I have y equals x to the 4th graphed as well. I'd like you to take a second and compare the overall behavior of x to the 4th to x squared. I know that all three of these graphs actually go through the origin. Which makes sense, because zero taken to any power is just equal to zero. So we plug in zero for x, and any of them, the y value is going to be zero as well. However, both of the graphs that have even powers have that property that I talked about in the previous answer video. Either end of the graph is going to point in the same direction. Either going to have a sort of U-shape overall or sort of upside down U-shape. The U for x to the 4th just happens to be a bit steeper than it does for x squared. So, let's see if this pattern continues as we move even higher in degree with our polynomial functions. Yet again, I've added more graphs. One of these is the graph of y equals x to the 5th, and one of them is the graph of y equals x to the 6th. So, thinking about the patterns that you noticed over here with our first three graphs that you're considering, what do you think the overall behavior of this 5th degree function will look like versus the 6th degree function?
This first graph over here, this green one, is the graph of y equals x to the fact, continue that pattern that we had started to pick up on over here. Just like x squared and x to the 4th, x to the 6this shaped sort of like a U. Although it's even steeper or skinnier, than the x to the 4th function. And it's base looks to be a bit wider. So, as you move away from 0, the graph isn't initially increasing very quickly. And then it shoots up really fast as soon as we move past 1. Think about why this is y. You might want to make a T chart with values at the input for the function smaller than 1, and see how the value of y increases as x increases. Now, just like y equals x to the 3rd, y equals x to the 5th has one tail, or one end of the graph pointing to negative infinity, and the other pointing to positive infinity. Then we can see that y equals x to the has pretty much the same shape. This is really interesting.
On these two separate graphs right now, I put all of the parent polynomial functions that we've talked about so far. And they're grouped according to which ones he said had similar end behavior. Or in other words, similar behavior at either tail of the graph moving away from to origin. However, I have added one extra graph, we didn't talk about in this session explicitly and that's the graph of y equals x. Remember that technically the variable x is a polynomial. It's just a polynomial of first degree. Pretty simple. I know that it looks pretty different from both x cubed and x to the 5th, but I do have a reason for putting it over here. You'll see in a second. First of all though, the similar behavior of graphs over here and of graphs over here, has led these groups of functions to have names. All the functions on one of these two graphs are even functions. And the other set are odd functions. So which one do you think is which? I've labeled this set of graphs a, and this set of graphs b. Just use your common sense to answer this question.
The functions on graph A are even functions and the functions on graph B are odd functions. This correlates, as we can see, with the powers that each of these functions have. Over here, our powers are 2, 4, and 6, and over here, they're 1,
I'd like to take a second to talk about a pretty important concept in Math in general, especially when we're talking about graphing. And that is symmetry. We talked about this to a certain extent when we were dealing with quadratic functions, like y equals x squared. We said that if you have a parabola like this one, we know that we can draw a line that goes straight through its vertex, such that, if you folded the problem in half along this line then it would map directly onto itself. In other words, at any point along this middle line, this axis of symmetry, the points on the parabola directly to the left or right of it, are equally distant from the axis. So if I'm at this position for example, this y value, then the distance in the extraction from here to here, is the same in terms of absolute value, as the distance from here to here. Now it seems looking at these other even functions, like a similar kind of symmetry applies to them. One question is, do odd functions have symmetry as well? And, in both cases, how do we mathematically express that symmetry. Now, I'm about to give you a kind of complex quiz that's going to require a fair amount of critical thinking and some new materials. So, this is going to definitely be a major challenge problem. So, if we know that a point, that we are going to call x comma y, is on a given graph. What if we find that the point negative x comma y is also on that graph? What does that say about this graph? Does that mean that is represent an even function, or an odd function, or neither of these? Think about which of the graphs either on this plane or this plane, this property applies to. And you can even test it out with real points.
This means that this is graphing an even function. Let's think about y. If we pick a y coordinate on any one of these graphs, let's say 10, [inaudible] distance we need to move to the right, in order to hit our curve, is the same as the distance we would need to move, to the left to hit the curve, just in the opposite direction of course. So that means the x coordinate, of the point whose y coordinate is 10, on the right side, is just the opposite of the x coordinate, of the point whose y coordinate is 10 to the left side of the y axis. This makes sense mathematically as well. Let's take a look at y equals x squared to see how. Let's say we have an x coordinate of 3, well then y is going to equal 3 squared, which is just 9. However we also know that if we had negative 3 instead Well, negative three squared is also nine. Negative numbers and positive numbers that the same absolute value, have the same squares.
Reflecting on what we learned about even functions in the last quiz, which of these descriptions do you think also apply to even functions? Do they have symmetry across the x-axis? Symmetry across the y-axis? Is it true for them that f of negative x is equal to f of x or that f of negative x is equal to negative f of x, please pick as many as you think are correct.
The line that runs down the middle dividing all of these even functions in half is the y axis. So they have symmetry across the y axis. Just by looking we can tell that they don't have symmetry across the x axis. In fact, none of these even functions even have any points that lie below the x axis. So that means that none of the points on them that lay above it, will have corresponding points that there reflected across from. Now, these last two choices might look a little bit complicated, but taking your time and thinking through them you can figure out which one's true, if either one. Saying that f of negative x Is equal to f of x, is basically saying that two points lie on a graph. This is the same as saying that if we have the point x, y, or x, f of x, then we'll also have the point negative x, y. This is saying that when you plug in the negative version of some x value, you get the same y value since the value of the function is the same at those two x coordinates. We already saw in the last quiz that this is in fact true of even functions. So this also applies. The last answer does not, since having it be true would mean that this third one couldn't be true.
So we've been focusing on even functions for the last few quizzes, and I think it's time that we pay attention to the odd ones for a little bit. So let's ask the same question of odd functions. Which of these choices apply in their case? Do they have symmetry across the x axis or the y axis? And do either of these rules about points on the functions apply?
If we try to fold any of the curves on this graph across the y-axis, none of them are going to map over to themselves. So, they must symmetry across the y-axis. The same is true of the x-axis when we try it there. Folding them in half along this line is not going to make points up here match points down here because they're on opposite sides of the y-axis. So, it looks like neither of these 2 symmetries applies in the case of odd functions. However, let's look at these last 2 choices. Maybe one of them works. We know that this is a property of even functions, that points equidistant from the y-axis have the same y value. But it doesn't look like this is true of odd functions. If I pick some x coordinate like 5, and I find the given y coordinate, then finding the opposite x coordinate, negative 5 does not give me the same y coordinate. It's all the way down here, instead of up here. However, these y coordinates are related. This one is the negative version of this one. So that means this last rule is true.
We just said that, for odd functions, f of negative x is equal to negative f of x. So I'm wondering if we can translate that to be a little bit more clear in terms of coordinates of points on the graphs of odd functions. So if we know that a point x, y lies on the graph of an odd function. Then what does this rule tell us is another point that lies on the function?
If we look at either side of this equation, we can figure out what points it corresponds to. Let's start off by remembering what this equation is saying. If we look at either side of this equation, we can translate each side into a point that lies on the graph of an odd function. This is part of the beauty of using function notation. F of negative x is the y value of a point. And it also tells us what the x value of that point is. This left side means then that we have the point negative x comma, let's call it y for a y coordinate. And the right side then, has x as its x coordinate since that's the input. And then its y coordinate, f of x, is just the negative version of this y coordinate. Now here's where it gets a little bit tricky. I didn't tell you that either of these points lies on the graph, I told you that point x comma y lies on the graph. What's the difference between this point and this point, though? Well, here we have negative x and then we have positive x. And here we have positive y and negative y. So it looks like for each coordinate, the only thing that has changes is the sign. If we do that to this point, then what we get is negative x coma negative y. So that's your answer, this is another point that lies in the curve.
Since we're talking about functions that are even, functions that are odd, and potentially functions that are neither ever nor odd, I've graphed a bunch of different curves over here for you. Curves and lines, and I've labeled each of them with a letter. I'd like you to tell me whether each of these graphs represents an even function, an odd function, or neither an even function nor an odd function.
Our even functions, those which are symmetric across the y axis, are A, B and F. This one, this one and this one. We have two odd functions in the graph. Graph C and graph D. And we have one function that is neither even nor odd, which is this graph E.
Now, what type of function is each graph representing? One of them is a constant function, one of them is a linear function, one a quadratic function, one an absolute value function, one a cubic function, and one a square root function. Please place the letter of the graph that goes with each of these types of functions in the box next to it.
Now you may not have heard the term constant function before, but intuitively we think that, this is a function that has a constant value, at all values of the input. That's what graph A is. It's at the form y equals some constant, C, let's say. Actually in this case it looks more like 3, so we'll just write that. Linear functions you recognize. It's graph C right here. The quadratic function is the parabola, graph F. The absolute value function, which you saw in a problem set earlier in the course, is graph B. The cubic function is graph D. And the square root function is graph E. Now you may not have seen this graph before either. But as always, you can test points picking values of x and then plugging them into the function to figure what the y coordinate should be. The graph of this function is y equals the square root of x. So if x equals 0, y equals 0. If x equals 1, y equals 1. And we can see that moving further over, if x equals 5, y equals just over 2.