WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about graphs.
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A **graph** is a visual representation of a function or equation.
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While perhaps not as precise as numbers and variables, a graph gives us an intuitive feel for how a function or equation works--how it looks.
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This graph is able to convey a wealth of information in a single picture.
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Now, just like functions, you have definitely been exposed to graphs by this point.
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You have seen them in previous math courses; but you might not have fully grasped their meaning.
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This lesson is going to crystallize our understanding of what is a graph is telling us about a function or an equation that it is representing.
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It will tell us what it means, exactly--that is what this lesson is here for.
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They get us all on the same base for graphs, so that we can move forward and understand everything that is going to come next.
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Graphs can tell us a whole bunch of information very quickly; they come up all the time in math.
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So, it is really, really important--we absolutely have to start by understanding what a graph represents,
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because we are going to see them all the time in math, and in sciences, and in other things.
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Having a really good understanding of what graphs mean is just going to matter for our understanding of a huge amount of other things.
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So, we really want to start on the right foot.
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All right, let's begin: when we have a graph, it shows how the input affects the output, or how one variable affects the other.
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But what does that mean, and how should we interpret the pictures we see?
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To answer that question, we are going to consider the graphs of f(x) = x + 1, and the equivalent graph of y = x + 1.
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This graph over here is the same for both of those--either that function, f(x) = x + 1, or that equation, y = x + 1.
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We are going to get that same graph on the right side.
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Remember from past math classes: we always associate the horizontal axis with the input independent variable.
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Our x is the input variable, the independent variable.
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And the vertical axis gets connected to the output, or the dependent, variable, which is normally going to be f(x), or y.
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So, over here, the vertical part connects to f(x), or y, while the independent input part connects to the x,
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for this function and this equation that we are going to be talking about.
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One way to think of a graph is as a way to see what happens to various inputs.
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If I plug in some number for x, where does it go? What happens to this number?
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The graph lets us see how different inputs are mapped to various outputs.
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We get to see a whole bunch of inputs getting mapped to a whole bunch of outputs, all at the same time; that is what a graph is showing us.
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So, let's interpret the graph of f(x) = x + 1 with this idea in mind.
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The reason (2,3) (2 is the horizontal, and 3 the vertical, portion) is on the line is because,
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if we use x = 2 as input, then if we plug in 2 into f, if we plug it in for that x, then we will get 2 + 1; and 2 + 1 is 3.
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So, if we plug in 2, it gives out 3 right here.
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That is where we are getting this graph from.
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This line that we see is all of these possible inputs on this x-axis.
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Each point on that line shows where the x-value directly below it is mapped.
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If we look at 7, then it tells us that that came out as an 8; if we look at -4, it says that that came out as -3.
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All right, we plug in a value from the x-axis, and it comes out on the y-axis; we get to see what this function does to that input value.
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And that is how we are looking at a graph: the input goes in from the horizontal, and the output comes out on the vertical, axis.
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It is a really great way of being able to see how the function affects many, many inputs,
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all at the same time, as opposed to having to look at a table where each one takes up its own entry;
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we can just see this nice curve, or this nice line, that explains many, many pieces of information very, very quickly and very, very succinctly.
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We can take it in in a single look.
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We can also think of it, though, as the location of solutions.
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This is another way to interpret the graph that is kind of different than that other one.
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They are connected, but they are also fairly different; and I think, in the way that we think about it, it has a really different meaning in our head.
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The graph of an equation is made up of all of the points that make the equation true.
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So, while that is the same thing as input to output in some ways,
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we are going to see that we can also say that the reason why this point is here,
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the reason why this point gets to be on our graph, is because it works with the equation; it is truth.
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The points that aren't on our graph, the points that aren't highlighted in the graph, but they are just on our plane--
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those don't make truth; those are false points, and so, since they would make the equation false, they don't get to be on the graph.
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Only the points that would make the equation true get to be on the graph.
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The graph is all of the truth points--all of the points that make our equation actually work.
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Let's interpret the graph of y = x + 1 with this idea in mind.
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The reason why (2,3) is on the graph...we go to (2,3)...the reason why this point here is on the graph
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is because, if we set that into our equation, (2,3), then if we plug that in, here is the 3; 3 is y;
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here is the x, which is 2; if we set that up as an equation, 3 = 2 + 1, yes, that is actually true; 3 does equal 2 + 1.
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So, because 3 equals 2 + 1, it is true; (2,3) gets to be on the graph, because the equation that would connect to that,
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3 = 2 + 1, is a true equation; every point on the line is a solution to the equation.
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It is all of the true points, all of the points that would make the equation true.
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8 = 7 + 1 gives us the point (7,8); -3 = -4 + 1 gives us the point (-4,3)...oops, not (-4,3), but (-4,-3); I'm sorry about that typo.
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And that is what is going on right there.
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If we were to put on some other point--let's just consider (0,10) for a second.
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We consider the point (0,10): if we were to plug that in, we would get 10 = 0 + 1.
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Wait, that is not true! 0 + 1 is not equal to 10; 10 does not equal 0 + 1, so this point here is a false point.
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It doesn't get to be on our graph; and that is why the graph is just made up of that red line.
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It is because those are all of the points that actually give us truth.
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If we went with some point that was not on that line, it would actually end up making our equation false; so it doesn't get to be on the graph.
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We can interpret the graph as the place of truth, the location of all of the solutions to the equation.
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This gives us two very different ways to interpret, and they are both totally valid and useful.
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That said, generally we are going to want to think in terms of the first one.
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Mostly, the first one is going to be the easier way to think about what a graph is telling us.
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For functions, it is almost always easiest to think in terms of how inputs are mapped to outputs.
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For equations, it is not always best; but we can normally use it, as well.
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We can normally use this method for equations, as long as they are in that form y = ....
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If it is set up with a bunch of y's showing up in multiple places, we can't really use this,
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because we do not have a good way to go from input to immediately showing us what the output has to be.
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So, it has to really be in this form, y = ...; but that is really what we are used to.
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When we see something like y = x² + 3x + 1, it is set up in this form of y = ...(things involving x).
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But in either case, as long as we are in this y = ..., or we are just looking at a straight function, f(x),
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in either of these two cases, this interpretation is a great way to think about graphs.
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We plug in an input, and then we get an output on the vertical.
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We plug in a horizontal location as the input, and that gives out a vertical location as the output,
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which gives us an ordered pair, which we can now plot on our plane, when all of those points put together make a graph.
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This is a really useful way; it is really easy to grasp; it is very intuitive; and it works very, very well.
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Still, at other times, it will actually be more useful to think in terms of solutions.
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What point is a solution? Where is it true?
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This idea is going to be especially important for certain types of equations that will get seen later on.
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But it is also going to matter for when we want to talk about
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where two equations or two functions intersect--where they have the same value at a certain point.
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That idea of where it is true--two things being true at the same time--that is an interesting idea,
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and useful for those locations, when we want to talk about intersection, or when we want to talk
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about certain more complex equations that are not just in the form y = ..., but where y shows up on both sides,
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or x and y are mixed up together...so sometimes we want to use that second form.
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But mostly, we want to think in terms of that first way.
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But the second way, we will occasionally use sometimes.
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Think in terms of that first way; think in terms of "input goes to output."
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But don't forget about the second way of "these are all of the places where it is true;
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these are all of the locations of the solutions," because sometimes we will need to switch gears and think in terms of that,
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because it will make things easier for us to understand at certain later points.
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All right, now that we understand what it is about, let's talk about axes.
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The **axes** are just the vertical axis and the horizontal axis--those lines that we are graphing on.
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The location of a graph can be as important as its shape.
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The location is set up by its axes; we want to pay attention to these axes.
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The axes will tell us where the graph is and what scale it has.
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Often, our axes are going to be square; that is to say, the x-axis is the same length as the y-axis.
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For example, we might have -10 to positive 10 on our y-axis, and -10 to positive 10 on our x-axis.
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This is a pretty common one; and this is square, because the x-axis is the same length as the y-axis.
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So, when we look at the picture, it is square, which is sort of an odd idea.
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But if we made it so that they had different lengths, but we had set them out as the same amount of line,
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then we would have a sort of squished picture; it wouldn't be the natural picture,
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where we think of width and length as meaning the exact same thing, in terms of length.
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That is a little confusing, because we are using the words width and length...I mean width and height meaning the same thing, and how long it is.
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So, as long as it is square, the graph isn't distorted from the square perspective we normally expect.
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However, sometimes it is going to be useful to graph functions on axes that are different from each other,
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where we are going to want to have a really, really big y-axis, but very small x-axis--
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where the function grows very, very, very quickly, so we want to be able to show all of its ability to grow.
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But since it does it so fast, we need a short x-axis.
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So, this is another really important reason to pay attention to the axes.
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You want to know how long they are, what amount of information is being represented in both of them,
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and also how big it is and where we are located.
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You want to have some sense of what the scale is: are they the same scale on both the x-axis and the y-axis?
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And just where are we located? Are we located in a weird place--is it not centered on 0?--those sorts of things.
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So, let's look at a single function: let's look at f(x) = x + 1 and see how many different graphs
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we can get out of it, just by changing the axes.
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Just by playing with the axes, we can get totally different-looking graphs.
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Here is the standard graph, our -10 to 10, -10 to 10.
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This top left graph here is basically what our standard graph would be.
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We are nice and square; the y-axis and the x-axis are the same length--that is what means to be square.
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It is from -10 to 10 and -10 to 10--numbers that we are used to and expecting.
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And also, the origin is in the center; we have (0,0) in the center of the graph.
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Now, let's consider the one below that--the bottom left.
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In this one, we have still square axes, because we are going from -2 to...actually...they are still square technically...-2 to 15 and -1 to 16.
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-2 to 15 means a length of 17; -1 to 16 means a length of 17; so even though they aren't putting down the exact same numbers,
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it is still a square, because they have the same length, total.
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-1 to 16 and -2 to 15 are both a length of 17; so it is still a square graph.
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This one here is square; this one here is square, as well.
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There is no distortion, no squishing in either the horizontal or in the vertical--no squishing of the graph.
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And the origin, though, is in a totally different place than the center of the graph.
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The origin is very, very bottom-left-corner; but it is still giving us the same x + 1.
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It looks kind of different, in terms of where the axes are; but it is still pretty clearly the same function making the graph.
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Let's look at another one: well, this one right here is actually not square.
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Why? Well, we have totally different lengths here: -10 to 10 and -5 to 5--
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that means the length of the horizontal is actually double the length of the vertical.
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The origin is still in the middle, so that is nice; that is something we are used to.
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But because we have a much shorter length, it ends up that we have more stuff in the horizontal than we do in the vertical.
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That means we have to compress what we are doing in the horizontal; so it has gotten squished left/right, which has caused it to stretch up vertically.
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This is not a square; these are not square axes right here.
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Another one that is not square (and hopefully you can read the yellow)...
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it is not too easy to read the yellow, but it is just me writing "not square" here.
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Once again, it is from -10 to positive 2 and -20 to something, but -10 to 2 is a length of 12, and -20 to something greater than 0 means greater than 20.
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So, once again, we have not-square axes; but this time, we have the vertical axis being longer than the horizontal axis.
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The horizontal axis has a length of 12; the vertical axis has a length of more than 20.
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So, that means that we have stretched it in the horizontal; as opposed to being squished horizontally,
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it has been stretched horizontally, because now it has less stuff to have horizontally than vertically.
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We have squished it vertically, because we are trying to cram in more vertical information while not having to cram in as much horizontal information.
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It has been squished vertically; so we have very different things here--vertical squish has happened in the bottom right one;
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and in the top one, we have horizontal squish, but it is not because of anything that has happened to the function.
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The function is x + 1 for every single one of these graphs; but the squish can be caused based purely on how we set up the axes.
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Setting up the axes, paying close attention to what the axes are telling us, is really important for us to actually understand what is going on in a function.
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Unless we understand what the axes are telling us, we won't actually know what this picture means.
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So, make sure you pay attention to axes; otherwise you can have no idea where you are.
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You have to have a map before you can really make sense of what is going on, and the axes are the map that our graph lives on.
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All right, one thing you might have noticed by this point is that the graphs in this course,
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unlike this one to the right, do not have arrows on them.
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I mean these arrows up here: at some point in the past, you have probably had a teacher who required you to draw arrows on the ends of your graph.
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And that made sense; they were trying to get across a very specific point to you.
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They were trying to remind you that the function keeps going on, even though we couldn't see it anymore.
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In a way, we can think of the axes as sort of boxing in the function.
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We don't get to see anything outside of the box of our axes.
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But in reality, the function doesn't stop at 3; it doesn't stop at -3, necessarily; this is just a nice, normal parabola.
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The function would keep going on; it would just continue off and off and off, and continue off and off and off and off.
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It doesn't actually stop; so the reason those arrows were there is to remind us that it goes past the edge of our axes.
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Just because the axes are here doesn't mean it stops; it is going to keep going.
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So, that is what those arrows were for; at this point, though, I think you have probably gotten used to that idea.
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We are not going to be using arrows at the ends of our graph in this course.
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The ends of our graphs in this course are just going to stop on our graphs; but that doesn't mean that the function stops.
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We are going to assume that we are all aware that the graph keeps going.
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It doesn't stop once it hits the edge; it just keeps going, unless we have been very specifically told that the function stops at a certain location.
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So, the graph is only stopping because the edge of the graphing axes stop.
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It is the graphing axes that are stopping the function, not the function itself.
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The function continues past the edge of our axes, unless in a very specific case, where we are told that it stops at some place.
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So, when we see this lack of arrows, it doesn't mean that it stops; it just means that we have to remember that it keeps going past the edge.
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The only reason it stops is because it has hit this boundary at the edge of it.
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It is not stopping because it actually stops; it is not stopping because the function stops.
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It is just stopping because we are looking through a window.
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If you look out through a window, if you are in a house, and you look out through the window,
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you can't necessarily see everything to the left and everything to the right.
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You can only see what you are currently looking through in the window.
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You have to move how you are looking through the window, or move the location of the window
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(although that would require a sledgehammer, and is something no one that you live with is going to be very happy about)--
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you can move the location of the window and be able to see different things outside; but the window fixes what you can see.
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That is what the graphing axes are doing to us: they are fixing what we can see in space.
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We are not going to use arrows in this course, because we know that graphs have to keep going.
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We are just seeing a tiny window on a much larger function.
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That said, even though, in this course, we are not going to use arrows, and we are all aware of it at this point,
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I want to point out that there are some teachers out there, and some books,
00:19:08.800 --> 00:19:13.000
that will still use arrows, and will still require you to use arrows.
00:19:13.000 --> 00:19:21.800
So, just because I am here saying that you probably don't need to use them--you are probably used to them by this point--
00:19:21.800 --> 00:19:28.200
doesn't mean that your teacher, if you are taking another course of the same type somewhere else--
00:19:28.200 --> 00:19:30.200
that that teacher is going to be OK with it.
00:19:30.200 --> 00:19:33.100
So, make sure that, if you have another teacher, if you have somebody else
00:19:33.100 --> 00:19:36.500
who wants you to draw arrows--make sure you do what they are telling you to do.
00:19:36.500 --> 00:19:39.300
So, do what they say as long as you are in their class.
00:19:39.300 --> 00:19:42.700
For my class, you don't have to; we know what we are talking about.
00:19:42.700 --> 00:19:49.100
But in somebody else's class, they might still want you to draw arrows, so be aware of that.
00:19:49.100 --> 00:19:54.600
How do we actually graph? The easiest way to graph a function is by thinking in terms of that input-to-output.
00:19:54.600 --> 00:19:57.700
Remember, you put in a number, and it gives out a number.
00:19:57.700 --> 00:20:03.400
So, we choose a few x-values, and we figure out what y-values get mapped to those x-values, and then we plot those points.
00:20:03.400 --> 00:20:07.000
For example, consider f(x) = x + 1, the one we keep working with.
00:20:07.000 --> 00:20:15.600
If we plug in -2, that will give out -2 + 1, which is -1; so that gets us the point (-2,-1), right here.
00:20:15.600 --> 00:20:21.000
If we plug in -1, that gets us 0; so that gets us the point (-1,0) right here.
00:20:21.000 --> 00:20:26.500
If we plug in the point 0, then that gets us 1, 0 + 1, so that gets the point (0,1).
00:20:26.500 --> 00:20:34.800
If we plug in 1, 1 + 1...we get 2, so we get (1,2); if we plug in 2, 2 + 1...we get 3, so that gets us the point (2,3).
00:20:34.800 --> 00:20:38.800
And now we have a pretty clear idea: it is just a straight line; it is just going to keep going.
00:20:38.800 --> 00:20:48.300
So at this point, we could come along, and we could draw in a straight line that just keeps going through all of these points.
00:20:48.300 --> 00:20:55.000
And we know what is going on right here: we are able to figure out that these points tell us that that is what the shape of this graph is.
00:20:55.000 --> 00:20:58.500
We don't have to graph all of the points perfectly in between, because it is pretty obvious,
00:20:58.500 --> 00:21:03.500
at this point, that they would all just end up being on this graph, as well, if we were to keep going
00:21:03.500 --> 00:21:10.000
with finer and finer steps, and how often we would check to see where inputs went to outputs.
00:21:10.000 --> 00:21:15.300
However, straight lines are not necessarily the best way to connect all of our graphed points together.
00:21:15.300 --> 00:21:20.800
In many ways, graphing is like playing a mathematical game of Connect the Dots.
00:21:20.800 --> 00:21:25.800
But we don't necessarily want to connect with straight lines; we usually want to connect with curves.
00:21:25.800 --> 00:21:28.100
For example, let's consider f(x) = x².
00:21:28.100 --> 00:21:35.200
Once again, here is a table that shows us input locations going to output locations, making points.
00:21:35.200 --> 00:21:41.200
(-3,9), (-2,4),(-1,1), etc....we can see all of these points on this graph right now.
00:21:41.200 --> 00:21:44.500
But let's look at what happens if we were to connect it all with straight lines.
00:21:44.500 --> 00:21:47.200
If we connect with straight lines, we get this picture right here.
00:21:47.200 --> 00:21:54.400
And while it is not a terrible representation of a parabola, it is not a very great representation of a parabola.
00:21:54.400 --> 00:22:03.600
A real parabola has curves going on; it curves out; it curves out, as opposed to going out just in these straight, jagged lines.
00:22:03.600 --> 00:22:09.000
So, we want to remember this fact: curves are normally what is going to connect our points, not straight lines.
00:22:09.000 --> 00:22:16.300
The real f(x) = x² is based on curves, so it looks like this picture right here.
00:22:16.300 --> 00:22:21.400
It is based on these nice, smooth curves connecting all of these points together.
00:22:21.400 --> 00:22:26.100
What about the fact that curves in one function are not necessarily going to look exactly like the curves in the next function?
00:22:26.100 --> 00:22:32.600
That is true, but mostly, the graphs of functions are smooth; we want to connect points to each other through smooth curves.
00:22:32.600 --> 00:22:38.500
So, whenever you are drawing a graph, make sure you are connecting things smoothly, without jagged, harsh connections.
00:22:38.500 --> 00:22:41.300
Each function is going to curve in different ways.
00:22:41.300 --> 00:22:45.600
Remember, the shape of a curve will be different: if we are using x², x² is going to give us
00:22:45.600 --> 00:22:50.400
a totally different curve...well, not totally different, but it will be slightly different than x⁴,
00:22:50.400 --> 00:22:53.500
which is going to be different than the cube root of x.
00:22:53.500 --> 00:22:59.500
Each function that we graph will have a slightly different curve, or maybe a massively different curve.
00:22:59.500 --> 00:23:03.600
But over time, you are going to become more familiar with the shapes of various functions.
00:23:03.600 --> 00:23:07.000
As you graph more and more functions, as you see more and more functions,
00:23:07.000 --> 00:23:14.100
you are going to think, "Oh, x² should graph in this general way; √x should graph in this general way."
00:23:14.100 --> 00:23:20.400
"The cube root of x, the x⁵...all of these things have curves that are slightly different."
00:23:20.400 --> 00:23:24.000
It should curve a little faster, curve a little slower...those sorts of things.
00:23:24.000 --> 00:23:28.600
Your previous experience with functions helps immensely, so just pay attention and think back:
00:23:28.600 --> 00:23:31.700
when have I graphed something similar to what I am graphing right now?
00:23:31.700 --> 00:23:36.900
And use that information to help you graph what you are working on at the moment.
00:23:36.900 --> 00:23:43.000
Finally, the idea that more points make a more accurate graph: this is an important idea.
00:23:43.000 --> 00:23:48.000
The more points you plot before drawing in your curves, the more accurate the graph becomes.
00:23:48.000 --> 00:23:51.300
Each point on the graph is a piece of information.
00:23:51.300 --> 00:23:56.600
So, it makes sense that, the more information we use to make our graph, the more accurate the graph is going to become.
00:23:56.600 --> 00:23:59.500
If we use more information, it will improve our graph.
00:23:59.500 --> 00:24:10.900
Let's look at a specific example: Consider f(x) =...this complicated monster of a function, (x³ - 2x² - 7x + 2)/x² + 1.
00:24:10.900 --> 00:24:17.900
And we plot it with various step sizes: what I mean is how big of a jump we have between the various test points that we are setting up.
00:24:17.900 --> 00:24:24.500
We are going from -4 to 4; so we will start at -4, and then we will step forward by 2.
00:24:24.500 --> 00:24:30.100
That is what I mean by a step size of 2; don't worry--this is Δx; it means change in x,
00:24:30.100 --> 00:24:33.900
and it is just a way of saying how much we are changing x each time.
00:24:33.900 --> 00:24:42.700
So, if we step forward 2, if we go from -4 here to -2 here, and then to 0 here,
00:24:42.700 --> 00:24:47.700
and then to 2 here, and then to 4 here, we have stepped forward by 2 each time.
00:24:47.700 --> 00:24:51.700
And we can evaluate...I am not putting the table down here, because it is just kind of a pain
00:24:51.700 --> 00:24:54.400
for us to have to see all of the numbers that we are going to be going through soon.
00:24:54.400 --> 00:24:58.400
But if we evaluated each one of these things, we get the following vertical locations.
00:24:58.400 --> 00:25:08.100
-2 happens to be at 0; 0 happens to be at 2; 2 happens to be somewhere between -2 and -2.5; and so on, and so forth.
00:25:08.100 --> 00:25:12.500
So, what happens if we increase the step size? We don't really have a very good idea of what this thing looks like.
00:25:12.500 --> 00:25:29.100
It might go like this, but it could also go like this; it could maybe even do something crazy, like this.
00:25:29.100 --> 00:25:35.700
We don't really have a good idea of what those points mean, because we haven't strung enough of them together to get a very good idea.
00:25:35.700 --> 00:25:40.700
We are not used to this function, (x³ - 2x² - 7x + 2)/(x² + 1).
00:25:40.700 --> 00:25:47.400
This is an unusual function; we are not used to graphing things like this, so we don't have a really good sense of what it is going to look like.
00:25:47.400 --> 00:25:50.900
So, since we don't have a really good sense of what it is going to look like, we don't have the expectations;
00:25:50.900 --> 00:25:54.700
we need more points down before we are going to be able to have a good sense of where it is going.
00:25:54.700 --> 00:25:58.200
Let's consider a smaller step size--a step size of 1.
00:25:58.200 --> 00:26:02.600
Now, we go from -4 to -3, then -2, then -1, then 0, etc.
00:26:02.600 --> 00:26:05.400
Now, we are starting to get a better idea of what the curve of the function looks like.
00:26:05.400 --> 00:26:08.100
We are starting to think, "Well, now we are starting to see what is happening."
00:26:08.100 --> 00:26:14.100
There is still a little confusion; we are not really quite sure what happens between -2 and 1 horizontal locations.
00:26:14.100 --> 00:26:18.000
But we are starting to get a better idea; let's make it an even smaller step size.
00:26:18.000 --> 00:26:24.100
We are at .5; oh, now it is starting to come in much clearer--we can start to understand what is going on.
00:26:24.100 --> 00:26:27.700
We go with .2; oh, now we are really starting to see what it is.
00:26:27.700 --> 00:26:36.100
We now have a great idea; finally, we go to .01; now there are so many points down that it almost makes a continuous, smooth line.
00:26:36.100 --> 00:26:41.000
The only place where it isn't quite smooth is this section in the middle right here,
00:26:41.000 --> 00:26:46.500
where the function is changing so quickly that we can actually still see the space between these tiny points.
00:26:46.500 --> 00:26:53.700
But when it is not changing that fast, like most of it here or here, we end up seeing that it strings together,
00:26:53.700 --> 00:26:58.300
because we have put down so many points that it basically turns into a smooth line.
00:26:58.300 --> 00:27:01.100
And that is exactly what happens when we make a graph.
00:27:01.100 --> 00:27:06.100
We are putting down so many points that we are saying, "Oh, that is what the smooth line is that it is making."
00:27:06.100 --> 00:27:09.100
That is what is happening when you use a graphing calculator, actually.
00:27:09.100 --> 00:27:15.000
If you use a graphing calculator, the computer inside is basically saying, "Make a bunch of points."
00:27:15.000 --> 00:27:22.700
It is now doing the same sort of thing; it is doing tiny, tiny steps, and then it is just stringing them all together with straight lines.
00:27:22.700 --> 00:27:28.700
So, it makes a whole bunch of points, and then it just strings them together; and that is what we see in the end.
00:27:28.700 --> 00:27:33.000
The way that you graph something is: you just keep using more and more points if you need more information.
00:27:33.000 --> 00:27:36.500
If you have a pretty good sense of how it is going to curve, though, you just have to put down enough points
00:27:36.500 --> 00:27:42.600
so that you can then put in the curves, because you have already had the experience of working with that function before.
00:27:42.600 --> 00:27:47.600
All right, when we introduce the idea of a function, we discussed an important quality for functions.
00:27:47.600 --> 00:27:51.600
For a given input, a function cannot produce more than one output.
00:27:51.600 --> 00:27:58.700
So, for example, we said that if f(7) = -11, then it can't also be true that f(7) = 20.
00:27:58.700 --> 00:28:05.800
Then that means that f(7) equals two things at once; and we said that, when you put something into a function, it always puts out the same output.
00:28:05.800 --> 00:28:10.600
So, if we put in f(7) the first time, and it gets -11, then the second time, it has to give -11,
00:28:10.600 --> 00:28:14.100
and the third time it has to give -11, and the fourth time it has to give -11.
00:28:14.100 --> 00:28:18.200
It can't ever be the case that all of a sudden, things go crazy and it produces a different result.
00:28:18.200 --> 00:28:24.300
No, we can trust our function; we can trust our transformation, our process, our map, our machine--whatever analogy we want to use.
00:28:24.300 --> 00:28:30.200
We can trust the function to always give us the same output if we put in the same input.
00:28:30.200 --> 00:28:37.700
So, if f(7) = -11, it can't be the case that f(7) equals something else, as well--something different than -11.
00:28:37.700 --> 00:28:41.400
We can turn this idea into a thing that we can see in graphs.
00:28:41.400 --> 00:28:47.000
We call this idea the vertical line test, and it says that if a vertical line could intersect
00:28:47.000 --> 00:28:51.900
more than one point on a graph, it cannot be the graph of a function.
00:28:51.900 --> 00:28:58.800
So, if we have a vertical line, and we bring it along like this,
00:28:58.800 --> 00:29:08.600
if we put a vertical line on anything over here on the left, it ends up not being able to intersect at more than one point.
00:29:08.600 --> 00:29:16.000
No matter where we bring a vertical line down on this graph on the left, it ends up passing the vertical line test.
00:29:16.000 --> 00:29:31.700
This over here is a function; but if we deal with this one over here, pretty much any point we choose will end up hitting two points:
00:29:31.700 --> 00:29:40.000
this one and this one--this one and this one; if we put it over here, it fails to hit any, but that doesn't necessarily mean it passes.
00:29:40.000 --> 00:29:44.300
If we can do it at any place on the graph, even if there is only one place on the graph
00:29:44.300 --> 00:29:50.500
where a vertical line hits the graph twice, then that means it is not a function.
00:29:50.500 --> 00:29:58.500
If there is a vertical line that could intersect more than one point, it is not a function.
00:29:58.500 --> 00:30:07.900
A vertical line--if it is able to intersect more than one location on the graph, it is not the graph of a function.
00:30:07.900 --> 00:30:11.200
Why--why is this the case? Well, consider this.
00:30:11.200 --> 00:30:14.500
Every point on a graph tells us where the x-value below is met.
00:30:14.500 --> 00:30:21.200
The points on the graph are in the form (x,f(x)); the x that we put into the function, and the f(x),
00:30:21.200 --> 00:30:26.300
the thing that the function puts out for that x--input and output put together.
00:30:26.300 --> 00:30:31.200
So, for example, let's look at this graph: this is the graph of something like a square root function.
00:30:31.200 --> 00:30:44.000
If on this graph we see, at x = 1, that we get f(1) = 2, we go to 1 on the horizontal; we bring it up, and we get to 2 on the vertical.
00:30:44.000 --> 00:30:49.500
So, we get that f(1) = 2, which is coming from the fact that the point is (1,2).
00:30:49.500 --> 00:30:52.200
So, we put in an input, and we get the output of 2.
00:30:52.200 --> 00:30:56.700
But let's consider this other one: what if we had this graph instead?
00:30:56.700 --> 00:31:06.000
On this graph, at x = 1, we get (1,2) and (1,-2); that means, since it is a graph,
00:31:06.000 --> 00:31:12.600
that if it is the graph of a function, we have f(1) = 2 and f(1) = -2.
00:31:12.600 --> 00:31:17.600
But that is not possible--a function cannot give out two different things.
00:31:17.600 --> 00:31:25.200
We can't plug in 1 and get 2 and -2; if we plug in 1, it is not allowed to give out two different outputs.
00:31:25.200 --> 00:31:33.100
That means we can't be looking at the graph of a function, because when we plug in one number, it gives out two things; so it fails the vertical line test.
00:31:33.100 --> 00:31:38.200
This picture right here is not the graph of a function.
00:31:38.200 --> 00:31:41.900
Remember, the domain is the set of all inputs the function can accept.
00:31:41.900 --> 00:31:44.500
We talked about this when we first talked about functions.
00:31:44.500 --> 00:31:54.200
The domain is the set of all inputs that a function can accept; the domain is what the function can act on--the numbers that the function can do something to.
00:31:54.200 --> 00:32:01.200
A graph shows where a function goes, so it means that we can see the domain in the graph.
00:32:01.200 --> 00:32:06.100
Every point on the x-axis that the graph is above or below is in the domain.
00:32:06.100 --> 00:32:13.500
So, every point on the x-axis that the graph is above or below has to be in the domain of that function.
00:32:13.500 --> 00:32:19.100
However, if we can draw a line on an x-value, and it does not cross the graph, then that x is not in the domain.
00:32:19.100 --> 00:32:30.800
A really quick example: if we had √x like this, then if we have tried drawing a vertical line here,
00:32:30.800 --> 00:32:37.700
that means that this horizontal location has to be in the domain, because it ends up having an output.
00:32:37.700 --> 00:32:42.800
If we plug in this horizontal, it comes out as this output; so that means that it must be in the domain.
00:32:42.800 --> 00:32:54.900
But if we go over here, this horizontal location never shows up in our graph, so it must be the case that it is not included in the domain.
00:32:54.900 --> 00:32:58.400
That horizontal location is not included in the domain.
00:32:58.400 --> 00:33:04.000
So, if you can draw a vertical line on an x-value, and it does not cross the graph, then that x is not in the domain.
00:33:04.000 --> 00:33:09.600
Remember, the domain is everything that the function can take in.
00:33:09.600 --> 00:33:19.300
So, if a graph is above a point, then that means it had to be able to take it in, because it gives out something over that horizontal location.
00:33:19.300 --> 00:33:23.300
This is a great way to visually notice the domain; but we have to be careful to remember
00:33:23.300 --> 00:33:26.700
that our function probably continues past the edge of our viewing window.
00:33:26.700 --> 00:33:30.600
Remember the axes that we had there; so if we are going to use this idea,
00:33:30.600 --> 00:33:33.700
we have to remember that, just because it seems to stop,
00:33:33.700 --> 00:33:38.000
or we don't see anything past the edge of the axis, that doesn't mean that the domain stops there.
00:33:38.000 --> 00:33:44.500
We just need to remember that it might continue on; we have to have some sense for how it looks beyond the edge.
00:33:44.500 --> 00:33:48.400
We need to have some familiarity; we need to think, "Where would this keep going to?
00:33:48.400 --> 00:33:54.200
Would this keep picking up those points in its domain, or would it stop for some reason?"
00:33:54.200 --> 00:33:57.300
Range is the set of all possible outputs a function can have.
00:33:57.300 --> 00:33:59.800
We also talked about this when we first introduced functions.
00:33:59.800 --> 00:34:06.400
It is all the numbers that our function could possibly produce; so domain is what could go in; range is what can come out.
00:34:06.400 --> 00:34:09.700
Like the domain, we can see the range of a function in its graph.
00:34:09.700 --> 00:34:14.400
Every point on the y-axis that the graph is left or right of is in the range.
00:34:14.400 --> 00:34:20.500
However, if you can draw a horizontal line on a y-value, and it does not cross the graph, then that y is not in the range.
00:34:20.500 --> 00:34:26.500
So, for example, let's consider x²; x² looks something like this.
00:34:26.500 --> 00:34:33.900
So, if we go to this horizontal location, we would be able to eventually go up and hit it; so it is in the domain.
00:34:33.900 --> 00:34:43.400
Similarly, we can go to this vertical location, and if we cut horizontally, there must be some domain location that puts that out.
00:34:43.400 --> 00:34:49.100
Now, it turns out that there are actually two different domain locations that put that out; but that is OK.
00:34:49.100 --> 00:34:53.600
Multiple domain locations--multiple inputs--can give the same output.
00:34:53.600 --> 00:35:01.000
f(2²) is equal to (-2)²; that is perfectly fine...4 and 4.
00:35:01.000 --> 00:35:07.600
It is OK that the same input gives the same output; but the fact that there is some input that gives that output
00:35:07.600 --> 00:35:11.800
means that it must be in the range, because it can be an output.
00:35:11.800 --> 00:35:18.600
So, we go to any location on our vertical axis, and if we draw a horizontal line and it cuts the graph,
00:35:18.600 --> 00:35:24.200
then that must mean that there is something that can input and give that output.
00:35:24.200 --> 00:35:36.900
Any location that is directly left or right of a vertical location means that that vertical location is in the range; that location, that number, is in the range.
00:35:36.900 --> 00:35:45.200
If, on the other hand, we can draw a horizontal line on a vertical location, and it does not touch the graph--
00:35:45.200 --> 00:35:48.800
that would not touch x²--then that means it is not in the range.
00:35:48.800 --> 00:35:52.500
And that makes perfect sense: down here are the negative numbers.
00:35:52.500 --> 00:35:59.900
So, can x² give out negative numbers? No, it can't--there is no real number that we can plug in that will give out a negative number.
00:35:59.900 --> 00:36:02.900
So, since there is no number that we can plug in to give out a negative number,
00:36:02.900 --> 00:36:06.300
then that means that we can't output negative numbers, so they can't be in the range.
00:36:06.300 --> 00:36:10.900
So, the range does not include any negative numbers, which is why, when we draw a horizontal line
00:36:10.900 --> 00:36:13.900
in any of these negative numbers, it is not going to touch the graph,
00:36:13.900 --> 00:36:19.400
because there is nothing that can make an output that would give a negative number.
00:36:19.400 --> 00:36:24.800
Just like with the domain, we have to be careful to remember that our function probably continues past the edge of our viewing window.
00:36:24.800 --> 00:36:33.400
That viewing window is just what we are looking through; so it is possible that your range is going to keep going, because the graph is going to keep going.
00:36:33.400 --> 00:36:38.600
So, we have to have some feeling for how the function will look past the edges of what we are able to see.
00:36:38.600 --> 00:36:44.300
Beyond the edge of our viewing window, we need to have some sense of what is going to keep going on.
00:36:44.300 --> 00:36:48.300
If we have no idea, we need to expand our viewing window, so that we can have a better idea
00:36:48.300 --> 00:36:52.100
and see, "Oh, yes, that would keep going," or "No, that actually stops."
00:36:52.100 --> 00:36:56.400
Otherwise, we will not be able to figure out exactly where the range is.
00:36:56.400 --> 00:37:01.200
Graphing calculators are really useful; if you haven't already noticed, this is a great time to point out
00:37:01.200 --> 00:37:04.900
that there is an appendix to this course that is all about graphing calculators.
00:37:04.900 --> 00:37:11.000
So, if you go the very bottom, and look at the appendix, there is an appendix about graphing calculators.
00:37:11.000 --> 00:37:15.200
So, it might be at the end of the course, but that does not mean you should watch it last.
00:37:15.200 --> 00:37:18.900
Graphing calculators are really, really useful for doing math.
00:37:18.900 --> 00:37:23.900
And you can also use software for graphing on computers or tablets or phones.
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There might be just something you can download and put on a phone, if you have access to a smartphone.
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And you can just start doing graphs on that really quickly and easily.
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So, graphing calculators can be extremely helpful for getting a feel for how functions work.
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If you are planning on taking calculus at some point, I definitely would recommend getting a graphing calculator in the near future.
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You are almost certainly going to want a graphing calculator for calculus, and so it won't hurt to have it now in precalculus.
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Even if you are not going to continue in math, you might find one useful for taking this course right now,
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and maybe for other science courses that you are currently taking, or will take in the future.
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So, if you are interested in getting a graphing calculator (and I would recommend it if you can afford it--
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and even if you can't afford it, there are some alternatives that I am going to talk about that are free or extremely inexpensive)--
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check out the appendix on graphing calculators; we are going to talk about all about
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how you can use them, what they are good for, why you might want one,
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what are some recommendations, things to look for, and that sort of thing.
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So, check out the appendix; there is a whole lot of information on graphing calculators there.
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It is really useful, and you are probably in a position where it is going to be useful for you to have a graphing calculator,
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since you are taking this course, and there is a very good chance you will go on to take calculus.
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I would definitely recommend to get a graphing calculator if you can afford it.
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So, check out the appendix; there is lots of information there.
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All right, we are ready for our examples: first, we are going to graph something.
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Graph f(x) = x² - 3x + 1: we have done this before, but let's just see a quick reminder.
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We want to do this by plugging in points and getting outputs.
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So, we are going to plug in x's, and we will get f(x)'s out.
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We plug in...we are not quite sure what this looks like, so let's start with a simple number that we can be pretty sure is easy to do; let's plug in 0 first.
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If we plug in 0, we get 0² - 3(0) + 1; that gets us 1.
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If we plug in 1, then 1² - 3(1) + 1...well, 1² is 1, minus 3, plus 1...so we have 2 - 3; we have -1.
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Keep going; we plug in 2; that will be 2² - 3(2) + 1; 2² is 4, minus 3(2); that is 6, so we have 4 - 6 + 1.
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4 + 1 is 5; 5 - 6...we have -1, once again.
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Let's try going in the other direction as well: let's plug in -1.
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I am just going to start skipping directly to the numbers, because at this point, we should probably be able to do this in our heads,
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or be able to do it on paper on your own, I'm sure; so we will just speed things up.
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(-1)² - 3(-1) + 1...that will get us positive 1.
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We plug in -2: (-2)² - 3(-2) + 1; (-2)² gets us 4, minus 3(-2) gets us...
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we should be able to do it in our head...that is ironic for me to have said that; maybe that would be a good reason to write it out.
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So, (-1)² - 3(-1) + 1...and this is also a good lesson in "never just trust yourself to immediately be able to do things in your head."
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(-1)² gets us positive 1; minus 3(-1) gets us positive 3; plus 1 gets us 5.
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(-2)² - 3(-2) + 1...we have 4 + 6 + 1; we have 11.
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And if we go forward one more, at 3, we are going to see 3² - 3(3) + 1.
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We would get 9 - 9 + 1, so we would get positive 1.
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And one more: if we plug in 4, we would get 4² - 3(4) + 1, so 4² is 16, minus 3(4) is 12, plus 1.
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So, 16 - 12 is 4; with 4 plus 1, we get 5.
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All right, so we have a lot of information, but there is one thing that we might notice.
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We might say, "Parabolas need a bottom"; we are graphing a quadratic, and while we haven't formally talked about them,
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I am sure you have seen parabolas quite a few times by now.
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We plug in 1; we get -1; we plug in 2; we get -1; we might realize that that doesn't actually give us a bottom.
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That is going to give us sort of a flat bottom, so there is probably some point in between them that is even lower.
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So, we want to have some sense of where it is going; so let's actually plug in a number in between them.
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Let's plug in 1.5; if we plug in 1.5, f(1.5), we get -1.25; I will spare doing that here, but we would get -1.25.
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I will actually do it here; so we plug in 1.5, so 1.5² - 3(1.5) + 1...
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1.5²...when we put that into a calculator or do it by hand, we get 2.25 - 3(1.5)...we get - 4.5 + 1.
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So, we have 3.25 - 4.5; we get -1.25; great.
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All right, so at this point, we have actually found something that seems like it could be the bottom; and it turns out that it actually is precisely the bottom.
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But we don't know that technically; we haven't formally talked about it.
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But at least it gives us a sense of where this is going to be bottoming out.
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So now, let's actually set up our axes, and let's plot the thing.
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Now, this never gets that low; it only gets down to -1.25; so let's make the bottom of our axis not that long.
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So, we will go to -1, -2, because we never even reach -2; and we will go up 1, 2, 3, 4, 5, and it would keep going.
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But we are going to top out, so we will never actually end up seeing the number 11,
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because we can't make it up that high on these axes, if we are going to keep them at this reasonable size.
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And let's keep it square; so the distance from the origin to a vertical one will be the same as the distance from the origin to a horizontal one.
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So, this is approximately square; I am just roughly drawing it by hand, but it is pretty good.
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1, 2, 3, 4, -1, -2, -3...and I would keep going to the left, but we know that we are never going to even see that point,
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because -2 is already out of where we are going to be able to plot.
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So, let's just plot our points now: let's see, 0 is at 1; we have 0 at 1, so (0,1)--we have that point.
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Let's go to the left first; -1 manages to make it up to -5, and we are already going to be past the graph when we are going to -2; it is out here.
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We plug in 1, and we are going to be at -1; we plug in 2, and we are going to be at -1.
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Let's plug in the point in between them: 1.5 is going to be at -1.25, so it is just a little bit below.
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3 is going to be at positive 1, and 4 is going to be at 5; so we curve this out, because we know it is a parabola.
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So, we have some sense of how the curve looks.
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All right, and it would keep going on and out; and it just keeps going, past the edge of our axis.
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All right, and that is how we graph it; so this is pretty much how we can graph anything.
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Plot some points on a T-table; plot some points on x and f(x), input and output.
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Plot the points; figure out where they are going to go; then actually put them onto the graph.
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Set up points, I mean; and then plot them onto the graph, and then connect it with curves, depending on how we know that kind of graph gets put together.
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All right, this is the graph of f(x) = x³ + x² - 6x.
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Using it, we are going to estimate the values of f at -1.8, f at -1, f at 1, and f at 2.5.
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Then, we are going to also estimate the values where f(x) = 0; and then finally, we will estimate the values where f(x) = -3.
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So first, this part right here, f at -1.8...what we do is just go to -1, -1.8...well, that looks around here.
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So, we go up; that looks like -1.8; we go up here, and we are about here, so it looks to be a little above the 8, somewhere between the 8 and the 9.
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If that is the case, I would say that looks like around 8.3 to me, give or take.
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We are just estimating, so we don't have to be absolutely, perfectly precise.
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But I would say 8.5 is a pretty reasonable guess; 8.3 is probably a little closer, so let's go with 8.3.
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f(-1.8) is equal to...it looks like 8.3; it is an estimate--it says "estimate"; it is a graph--
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we are never going to be able to perfectly pull information from the graph.
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Well, we might be able to in a few cases; but it is going to be normally something
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where we are getting that we are pretty confident, but it might be slightly off by .01 or .01...
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well, that is the same thing...by .1, lower by .1, these sorts of things.
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It is hard to be absolutely, perfectly precise, since we are looking at a picture; but we can get a pretty good idea.
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The same thing for everything else: for f(-1), we just go to -1; we go up; f at -1 seems to be about this high.
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So, I would say probably about 5.7 or 5.6; so let's say it is 5.7.
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f at 1...we are here; we drop down; and that one looks like it is really pretty much exactly -4; so f(1) = -4.
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f(2.5)...plug in 2.5; we go up pretty high; that looks like it was a pretty good vertical...
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look, I would say that looks like it is pretty close to being right on 8; so we will say that that is 8; great.
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We have estimated the values for all of them; they might be a little bit inaccurate, but they are pretty close to right.
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And that is what a graph gets us--it gets us a good way to get a really good sense of what is going on.
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It might not be perfectly, absolutely, exactly right; but it will get us there pretty close, which is normally enough to be able to do stuff for lots of things.
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Now, let's look at estimating the x-values for f(x) = 0; we will do this one in blue.
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Estimate the x-value for f(x) = 0; so what is f(x) = 0?
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Well, remember, the vertical axis is f(x); that is the output.
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So, if that is the case, then we are looking for everything that is at the 0 height, which is the same thing as the x-axis.
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If it is crossing the x-axis...it looks like here like it crosses the x-axis at 2, crosses the x-axis at 0, and crosses the x-axis at -3 precisely.
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There is nothing else crossing there; so we can assume that we have found all of the x-values.
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It seems that is going to be x = -3, 0, and 2; they all caused f(x) to come out as 0; great.
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Finally, we will use red for the very last one; hopefully, that won't be too confusing.
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Estimate the x-values for f(x) = -3; if that is the case, we go to where f(x) is -3.
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f(x) is -3; we want to go and see...here is something; here is something; and that is pretty close to horizontal--not perfect; sorry.
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And there is something; so f(x) = -3 at these three horizontal locations.
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So, once again, it is not absolutely, precisely, absolutely perfect, but pretty good.
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f(x) = -3 is going to be at the x's that are...the first one, I would say, is a little past -3, but not by much, so probably -3.2.
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And then, the next one looks like it is around...just a little past where the...here is positive 1; that is right here, so this is 0.5.
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That is right here, so I would say that is just a hair past 0.5, so let's say that is 0.6.
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And then finally, here it is just a little past 1.5; I would say it is a little bit more past it, though; so probably 1.6, or maybe 1.65.
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Let's go with...let's say 1.7; maybe 1.6, maybe 1.7; but it is a little past 1.5, and we are sure of that.
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That is how we use a graph to figure out things from it.
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We can estimate values given an input, or we can estimate values given an output.
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We figure out what makes that output or where that input would get mapped to.
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What would that input get output as?
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Vertical line test: Which of the below is not the graph of a function?
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This one is not too hard: if the v's are the entirety that we are seeing, we just have to use the vertical line test.
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If we come along this one, and we put a vertical line on this one, it is pretty easy to see that it is not going to fail at any point.
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The vertical line is never going to cross it at anything.
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The only place where you might be a little curious is right here where it curves up.
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But it never really continues on in such a way that we can be sure.
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Any vertical line that we are making seems to cut it just once.
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Now, it does have this part where it sort of curves like this, but that is inaccurate.
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It looks like that, but the graph is actually curving a little more like this.
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And the reason why it looks like it is stacked on top of itself is because we have to add thickness to our line.
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In reality, the line is actually thinner than that; and is even thinner than that, because a point is infinitely thin.
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So, there is no stacking, because of that infinite thinness; it is only because of that thickness of our line
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that it ends up looking like there is something stacked.
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So in reality, if we come along with the vertical line test, since the vertical line is also infinitely thin,
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it is not going to cut it twice, because it doesn't really curve back on itself; it is only going to hit one thing at one point.
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So, this is a function.
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What about this one right here? This one is really easy to see that it fails.
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If we cut in the middle, it is going to hit a bunch of times.
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It cuts here, here, and here; so that is more than 1 intersection.
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If we go on the far sides, it will pass; but all we need is one place of failure, once place where cuts across multiple times.
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So, in the middle, it manages to fail being a function, because one input manages to simultaneously have three outputs.
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So, it is not a function.
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Finally, this one over here is the same idea as the left side.
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Even though it looks like it is starting to get vertical, it is never actually vertical at any point.
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It just needs to be an infinitely thin line to really understand what is going on, and a vertical line has to be infinitely thin, as well.
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So, we have to think about this, beyond just saying, "Well, it looks kind of stacked, so it must be."
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No, we have to think, "Oh, that is really only approximating where the graph is, because the line,
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while we can't see infinitely thin things...that is what the line is representing."
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So, it is the case that this one is also a function, because there is nothing where it clearly cuts two places at once.
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Great; that is how we use the vertical line test.
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Just drop vertical lines, and if there is any place where it clearly cuts the graph more than once, then it is not a function.
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If we can drop vertical lines everywhere, and it would never cut the function more than once, then it is a function.
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Final example: Prove that there is no function that could produce a circle as its graph.
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This might seem a little complicated at first; so what we want to do is think, "Well, how could we prove this?"
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Well, if we want to prove it, we need to show something involving circles as graphs.
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We get stuck too much on trying to think, "What is the right way to do this?"
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We might never get any progress.
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But if we think, "Well, what does a circle look like?" a circle has things stacked--it would fail the vertical line test.
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So, we know we can prove this by contradiction.
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Proof by contradiction: we are going to start by assuming that there is such a thing.
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So, proof by contradiction: assume such a function exists.
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If there is a function that could produce a circle, then look at its graph.
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Since it is a circle, we know what the graph of a circle looks like; who knows where it is going to show up on the graph, but we know it has to show up somewhere.
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So, here is a circle; and while it is not a perfect circle--I am but a mortal--it is a good idea.
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We can say, "Look, just take this and cut it at any place; any place inside of the circle, we are going to fail the vertical line test."
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The graph must fail the vertical line test; therefore, it is not a function--it cannot be a function.
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The graph cannot be a function; but it was the graph of a function.
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So, since the graph cannot be a function, it must be that no such function exists.
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So, our assumption was that the function did exist; since the graph cannot be a function,
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but it was just the graph of a function, then there is a contradiction.
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The function cannot exist, so it must be that no such function exists; and we are done--that is our proof.
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All right, assume that what we can see on the graph below is the entirety of the function f.
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In other words, there is nothing past the edge of the axes.
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We are looking through that window, but we have been told that there is nothing interesting past the edges of the window.
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So, this graph here is the entirety of the function f.
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Now, we want to estimate the domain and range of f from the graph.
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Now remember, the domain was everything that can be input.
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So, if we go to, say, 0, look: 0 shows up in the graph.
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Well, what about -3? -3 never shows up in the graph; there is nothing that it gets graphed to--nothing that it gets output as.
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It looks like the edge is -2; it looks like -2 is the very edge; and over here, 3 gets put in; 4 doesn't get put in;
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but it looks, probably, like 3.5 gets put in, so we would say that the domain is going to go from -2 to 3.5.
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What about the range? Range is everything that can be output.
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Is there anything that can output at 1? Yes, 1 manages to touch here, and manages to touch here.
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There is some input that gives out 1; if we put in an input here, we can see that it connects here.
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But if we go to 3 and we cut across, 3 horizontally never touches the graph, so it must be the case that there is no input that produces 3.
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So, 3 is not in the range; the highest that we manage to get to is right here.
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So, it looks like 1.5 is the highest that we managed to get to with the graph.
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It never shows up over here, but that is OK, because it shows up somewhere.
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And then finally, it looks like the lowest we manage to get to is -2.
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So, our range: the lowest location on our range is -2, and the highest location that we manage to make it to is 1.5.
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And we hit everything in between: if you go to any higher location in between, it shows up.
00:57:57.400 --> 00:58:04.600
So, our range is everything in between -2 and 1.5, because all of them have something that they are able to contact; great.
00:58:04.600 --> 00:58:09.400
All right, I hope you have understood what is going on here; I hope it has really crystallized the idea of a graph.
00:58:09.400 --> 00:58:12.900
Graphs are so important; they are going to show up in so many things in math.
00:58:12.900 --> 00:58:16.100
And they are also going to show up in science, and even if you just look in a newspaper.
00:58:16.100 --> 00:58:18.700
Graphs make up a really, really big part of mathematics,
00:58:18.700 --> 00:58:21.700
so it is really important that we understand what is going on with them now,
00:58:21.700 --> 00:58:23.600
because we are going to see a lot of them as we go on.
00:58:23.600 --> 00:58:26.000
All right, we will see you at Educator.com later--goodbye!