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### Graphs

• A graph visually represents a function or equation in math. It gives us an intuitive picture of how the function "works".
• There are two main ways to interpret what a graph means:
• Input ⇒ Output: The graph tells us what happens to each input value. "If I plug in some number for x, where will it go?" The input values are on the horizontal, the outputs are on the vertical.
• Location of Solutions: We can also interpret a graph as the location of all solutions to the equation. The graph of an equation is made up of all the points that make the equation true.
Between these two options, it's usually best to interpret it as the first one: how inputs are mapped to outputs. This gives us an intuitive way to see what happens as we change input values. However, the other way will occasionally come in handy, so don't forget about it entirely.
• Pay attention to the axes! The axes tell you where the graph is and what scale it has. Knowing this is important if you want to interpret what the graph means. [This is also called the graphing window.]
• In this course, we will not put arrows on the ends of our graphs. Instead, we assume we're all aware the graph keeps "going" past the edge. We won't use arrows because we know that most graphs are just a tiny window on a much larger function. [Caution: Some teachers might still want you draw arrows on the ends of your graphs. If that's the case, do what they say as long as you're in their class.]
• The easiest way to plot graphs is to plot points one-by-one. Make a table of values, calculate various inputs and outputs, then plot them on the graph. Once you have enough points to see the shape, draw it in.
• Almost always, the plotted points will connect with curves. As you see more and more functions, you'll start to learn the various shapes. Use this knowledge to help you draw graphs accurately.
• Anytime you're not sure how to draw in a graph, just plot more points. As you plot more points, you have more information. As you have more information, the picture becomes easier to see. This is always an option, even for the most confusing graphs.
• We can tell if a graph is the graph of a function with the Vertical Line Test. If a vertical line can be drawn that crosses the graph at more than one point, it is not a function. Why? Because this means a single input is mapped to two outputs, so it can't be a function.
• The domain of a function is all the inputs that a function can accept. Thus, every point on the x-axis that the graph is above or below is in the domain. However, 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. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
• The range of a function is all the possible outputs a function can create. Thus, every point on the y-axis that the graph is left or right of is in the range. 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. [Be careful to remember that our function probably continues past the edge of our "viewing window", so we need to have a sense for what happens beyond the edge.]
• If you haven't already noticed it, this is a great time to point out that this course has an appendix that's all about graphing calculators. Check out the appendix to learn more about graphing calculators, where you can find some free options, what they're good for, and how to use them.

### Graphs

What are the two main ways to interpret what a graph represents? Of those two ways, how do you normally want to interpret a graph?
• The most common way to make a graph is by creating a table of input values and corresponding output values. This is because a graph shows how input is transformed to output: given some x as input, what f(x) or y do we get as output?
• Alternatively, if we take any point on a graph and plug it in to the equation that created it, the equation will be true. Thus, every point on a graph is a solution to the equation: where the equation is true.
• In general, the first way is more useful for creating graphs and for understanding how they work. While the second way has its uses, it is usually more helpful to think in terms of input ⇒ output.
The two ways to interpret a graph are 1) How input (x-axis) is mapped to output (y-axis); 2) the location of all solutions to the equation (where it is true). Normally you want to think in terms of input ⇒ output.
Consider this graph of a function. Why does the graph stop at the edge of the graphing axes? [The space created by the axes for graphing is sometimes called the `graphing window'.] What happens beyond that edge?
• For most functions, the function goes on forever. We can put in arbitrarily large numbers, but how could we possibly create a graph that shows every single number? We have to stop somewhere, so we usually stop at the edge of the axes.
• However, once again, most functions keep going beyond the edge of the axes. We had to stop because we can't draw forever, but the function keeps going past the edge of the graphing window.
• Just because we see only a portion of the graph does not mean the function stops at that edge (unless we are explicitly told it does stop). It is up to us to have a sense for how it behaves beyond that edge if we need to know what happens outside the graphing window.
The graph stops at the edge of the graphing axes (graphing window) because it has to stop somewhere. Most functions go on forever, so the graphing window is just a small window on a much larger thing. Beyond the edge, the graph continues on (unless we are explicitly told otherwise).
The following table of values is for the function f(x) = 2x2−8. Using the table of values, graph the function.
 x
 f(x)
 −3
 10
 −2
 0
 −1
 −6
 0
 −8
 1
 −6
 2
 0
 3
 10
• Begin by choosing an appropriate scale for the axes by considering the values in the table. The most extreme x-values are −3 and 3, so a scale of x:[−4, 4] would probably be good. The most extreme y-values are −8 and 10, so a scale of y:[−10,10] would probably be good (that way we have the same amount above and below the x-axis).
• Once the axes are set up, plot all the points. Remember, we interpret the table as a bunch of ordered points ( x,  f(x) ). The first coordinate is plotted on the horizontal axis, and the second coordinate is plotted on the vertical axis.
• Connect the points with curves. Notice how the vertical change increases more and more the farther we get away from x=0. Thus, the curves around x=0 are more gently sloped, while they become steeper as they get farther out.
Graph the function f(x) = 0.5x + 2.
• Begin by creating a table of values. You could choose any x values to use, but it would probably be best to keep them small and near 0, since we probably want the graph to be centered around the origin.
• Creating the table of values, we get something like
 x
 f(x)
 −3
 0.5
 −2
 1
 −1
 1.5
 0
 2
 1
 2.5
 2
 3
 3
 3.5
Looking at the values, you might realize that you're graphing a line. You might also have realized it would be a line just from reading the function.
• Choose appropriate axes. In this case, square axes of [−5,5] on both sides would work nicely, but you could choose something else as well.
• Plot the points, connect them appropriately. This is particularly easy because the function is a line.
Graph the equation x2 +y = 5.
• First, we need to put the equation into a format that we can easily graph: y=stuff. Moving things around in the equation, we obtain:
 y = 5 − x2.
• Create a table of values. You could choose any x values to use, but it would probably be best to keep them small and near 0, since we probably want the graph to be centered around the origin.
• Creating the table of values, we get something like
 x
 y
 −4
 −11
 −3
 −4
 −2
 1
 −1
 4
 0
 5
 1
 4
 2
 1
 3
 −4
 4
 −11
From the values and the function, we realize we are graphing a parabola. Knowing this will help us curve the graph appropriately.
• Choose appropriate axes. x:[−5,5] and y:[−10,10] would work well for this, although you could choose different axes.
• Plot the points, connect them appropriately. Notice that (−4,  −11) and (4,  −11) will be just outside the bounds of the graphing window. That's okay. You can plot these points just past the edge and have the graph pass out of the window, or when drawing the graph, you can just know what the curve is heading towards.
Graph the function f(x) = √{|x+2|} − 2.
• Begin by noticing that the function has an absolute value wrapped in a square root. If you forgot that the vertical bars |  | represent absolute value, you'd want to begin by looking up what they mean. You'd find out that it turns whatever is inside of them positive.
• Create a table of values. You could choose any x values to use, but it would probably be best to keep them near 0, since we probably want the graph to be centered around the origin. Furthermore, we will get the "nicest" points when the value inside of the square root lines up with a perfect square. You might want to choose your x-values accordingly.
• Creating the table of values, we get something like
 x
 f(x)
 −18
 2
 −11
 1
 −6
 0
 −3
 −1
 −2
 −2
 −1
 −1
 2
 0
 7
 1
 14
 2
From the values and the function, we realize we are graphing something with the shape of a square root on both sides. Knowing this will help us curve the graph appropriately.
• Choose appropriate axes. Since the x-values get so large, we probably want to choose something like x:[−20,20]. For the y-axis, we could choose lots of things, but let's go with y:[−10,10] to show off how little height the function gains.
• Plot the points, connect them appropriately. Notice that our last points don't make it the edge of the graphing window. That's okay. Just keep drawing the graph out with the correct curve. You have enough information from the points you've seen so far to know how it will continue to curve. [If you want more information though, you can always add entries to your table of values.]
Is it possible for this graph to be the graph of a function?
• To check if a graph could be the graph of a function, we use the Vertical Line Test.
• The Vertical Line Test says that if a vertical line could intersect more than one point on a graph, it can not be the graph of a function.
• If we place a vertical line almost anywhere on this graph, it clearly cuts the graph in more than one place. Thus, it fails the Vertical Line Test and cannot be a function. (This is because the function would need one input to be mapped to two outputs, which a function is not allowed to do.)
No, because it fails the Vertical Line Test.
Is it possible for this graph to be the graph of a function?
• To check if a graph could be the graph of a function, we use the Vertical Line Test.
• The Vertical Line Test says that if a vertical line could intersect more than one point on a graph, it can not be the graph of a function.
• While it may be hard to see at first, if we place a vertical line anywhere on this graph, it only cuts the graph once. There is no place we could place a vertical line that would cut it twice, so it passes the Vertical Line Test, so it could be a function.
Yes, because it passes the Vertical Line Test.
Assume that what we can see on the graph is the entirety of the function f. (In other words, there is nothing past the "edge" of the axes.) Estimate the domain and range of f from the graph.
• The domain is all the inputs that the function can accept. Since every point on the graph that is above or below a given x-value accepted that as an input, those x-values make up the domain.
• Looking at the graph, the function accepts inputs of x:[−8,4]. Thus, the domain of the function is [−8, 4].
• The range is all the outputs that the function can create. Since every point on the graph that is left or right of a given y-value gave that as an output, those y-values make up the range.
• Looking at the graph, the function creates outputs of y:[−3,5]. Thus, the range of the function is [−3,5].
Domain: [−8,4];  Range: [−3,5]
Here is a graph of the function f(x) = x2+4x+1. Using the graph, determine the domain and range of the function.
• The domain is all the inputs that the function can accept. Since every point on the graph that is above or below a given x-value accepted that as an input, those x-values make up the domain. However, the domain can be larger than just what we see on the graph, since the graph is only a window on a larger function.
• We don't actually get much help from the graph for finding the domain here. Instead, we have to realize that the function can accept any x-value as an input and work fine. Alternatively, we can realize that if we were to keep "zooming out" the graph, it would keep expanding to the left and right forever.
• The range is all the outputs that the function can create. Since every point on the graph that is left or right of a given y-value gave that as an output, those y-values make up the range. However, the range can be larger than just what we see on the graph, since the graph is only a window on a larger function.
• The graph helps us partially in figuring out the range. The graph lets us see the lowest value produced by the function: −3. However, to find the whole range, we have to realize that the function can keep going up forever. If we "zoom out" on the graph, it would continue up forever.
Domain: (−∞, ∞);  Range: [−3, ∞)

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Graphs

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• Introduction 0:04
• How to Interpret Graphs 1:17
• Input / Independent Variable
• Output / Dependent Variable
• Graph as Input ⇒ Output 2:23
• One Way to Think of a Graph: See What Happened to Various Inputs
• Example
• Graph as Location of Solution 4:20
• A Way to See Solutions
• Example
• Which Way Should We Interpret? 7:13
• Easiest to Think In Terms of How Inputs Are Mapped to Outputs
• Sometimes It's Easier to Think In Terms of Solutions
• Pay Attention to Axes 9:50
• Axes Tell Where the Graph Is and What Scale It Has
• Often, The Axes Will Be Square
• Example
• Arrows or No Arrows? 16:07
• Will Not Use Arrows at the End of Our Graphs
• Graph Stops Because It Hits the Edge of the Graphing Axes, Not Because the Function Stops
• How to Graph 19:47
• Plot Points
• Connect with Curves
• If You Connect with Straight Lines
• Graphs of Functions are Smooth
• More Points ⇒ More Accurate
• Vertical Line Test 27:44
• If a Vertical Line Could Intersect More Than One Point On a Graph, It Can Not Be the Graph of a Function
• Every Point on a Graph Tells Us Where the x-Value Below is Mapped
• Domain in Graphs 31:37
• The Domain is the Set of All Inputs That a Function Can Accept
• Be Aware That Our Function Probably Continues Past the Edge of Our 'Viewing Window'
• Range in Graphs 33:53
• Graphing Calculators: Check the Appendix! 36:55
• Example 1 38:37
• Example 2 45:19
• Example 3 50:41
• Example 4 53:28
• Example 5 55:50

### Transcription: Graphs

Hi--welcome back to Educator.com.0000

Today, we are going to talk about graphs.0002

A graph is a visual representation of a function or equation.0005

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.0009

This graph is able to convey a wealth of information in a single picture.0018

Now, just like functions, you have definitely been exposed to graphs by this point.0022

You have seen them in previous math courses; but you might not have fully grasped their meaning.0027

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.0032

It will tell us what it means, exactly--that is what this lesson is here for.0039

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.0043

Graphs can tell us a whole bunch of information very quickly; they come up all the time in math.0048

So, it is really, really important--we absolutely have to start by understanding what a graph represents,0054

because we are going to see them all the time in math, and in sciences, and in other things.0062

Having a really good understanding of what graphs mean is just going to matter for our understanding of a huge amount of other things.0066

So, we really want to start on the right foot.0073

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.0075

But what does that mean, and how should we interpret the pictures we see?0085

To answer that question, we are going to consider the graphs of f(x) = x + 1, and the equivalent graph of y = x + 1.0089

This graph over here is the same for both of those--either that function, f(x) = x + 1, or that equation, y = x + 1.0096

We are going to get that same graph on the right side.0104

Remember from past math classes: we always associate the horizontal axis with the input independent variable.0106

Our x is the input variable, the independent variable.0115

And the vertical axis gets connected to the output, or the dependent, variable, which is normally going to be f(x), or y.0119

So, over here, the vertical part connects to f(x), or y, while the independent input part connects to the x,0129

for this function and this equation that we are going to be talking about.0139

One way to think of a graph is as a way to see what happens to various inputs.0144

If I plug in some number for x, where does it go? What happens to this number?0148

The graph lets us see how different inputs are mapped to various outputs.0154

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.0160

So, let's interpret the graph of f(x) = x + 1 with this idea in mind.0167

The reason (2,3) (2 is the horizontal, and 3 the vertical, portion) is on the line is because,0171

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 3.0181

So, if we plug in 2, it gives out 3 right here.0192

That is where we are getting this graph from.0199

This line that we see is all of these possible inputs on this x-axis.0201

Each point on that line shows where the x-value directly below it is mapped.0205

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 -3.0209

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.0227

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.0235

It is a really great way of being able to see how the function affects many, many inputs,0242

all at the same time, as opposed to having to look at a table where each one takes up its own entry;0246

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.0251

We can take it in in a single look.0257

We can also think of it, though, as the location of solutions.0260

This is another way to interpret the graph that is kind of different than that other one.0263

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.0267

The graph of an equation is made up of all of the points that make the equation true.0276

So, while that is the same thing as input to output in some ways,0280

we are going to see that we can also say that the reason why this point is here,0285

the reason why this point gets to be on our graph, is because it works with the equation; it is truth.0289

The points that aren't on our graph, the points that aren't highlighted in the graph, but they are just on our plane--0297

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.0302

Only the points that would make the equation true get to be on the graph.0309

The graph is all of the truth points--all of the points that make our equation actually work.0312

Let's interpret the graph of y = x + 1 with this idea in mind.0320

The reason why (2,3) is on the graph...we go to (2,3)...the reason why this point here is on the graph0324

is because, if we set that into our equation, (2,3), then if we plug that in, here is the 3; 3 is y;0332

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 + 1.0341

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,0349

3 = 2 + 1, is a true equation; every point on the line is a solution to the equation.0359

It is all of the true points, all of the points that would make the equation true.0365

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.0369

And that is what is going on right there.0385

If we were to put on some other point--let's just consider (0,10) for a second.0388

We consider the point (0,10): if we were to plug that in, we would get 10 = 0 + 1.0394

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.0402

It doesn't get to be on our graph; and that is why the graph is just made up of that red line.0410

It is because those are all of the points that actually give us truth.0417

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.0420

We can interpret the graph as the place of truth, the location of all of the solutions to the equation.0426

This gives us two very different ways to interpret, and they are both totally valid and useful.0433

That said, generally we are going to want to think in terms of the first one.0439

Mostly, the first one is going to be the easier way to think about what a graph is telling us.0443

For functions, it is almost always easiest to think in terms of how inputs are mapped to outputs.0448

For equations, it is not always best; but we can normally use it, as well.0453

We can normally use this method for equations, as long as they are in that form y = ....0456

If it is set up with a bunch of y's showing up in multiple places, we can't really use this,0462

because we do not have a good way to go from input to immediately showing us what the output has to be.0466

So, it has to really be in this form, y = ...; but that is really what we are used to.0472

When we see something like y = x2 + 3x + 1, it is set up in this form of y = ...(things involving x).0476

But in either case, as long as we are in this y = ..., or we are just looking at a straight function, f(x),0485

in either of these two cases, this interpretation is a great way to think about graphs.0491

We plug in an input, and then we get an output on the vertical.0495

We plug in a horizontal location as the input, and that gives out a vertical location as the output,0499

which gives us an ordered pair, which we can now plot on our plane, when all of those points put together make a graph.0505

This is a really useful way; it is really easy to grasp; it is very intuitive; and it works very, very well.0513

Still, at other times, it will actually be more useful to think in terms of solutions.0519

What point is a solution? Where is it true?0523

This idea is going to be especially important for certain types of equations that will get seen later on.0527

But it is also going to matter for when we want to talk about0534

where two equations or two functions intersect--where they have the same value at a certain point.0536

That idea of where it is true--two things being true at the same time--that is an interesting idea,0543

and useful for those locations, when we want to talk about intersection, or when we want to talk0547

about certain more complex equations that are not just in the form y = ..., but where y shows up on both sides,0552

or x and y are mixed up together...so sometimes we want to use that second form.0559

But mostly, we want to think in terms of that first way.0564

But the second way, we will occasionally use sometimes.0567

Think in terms of that first way; think in terms of "input goes to output."0571

But don't forget about the second way of "these are all of the places where it is true;0575

these are all of the locations of the solutions," because sometimes we will need to switch gears and think in terms of that,0579

because it will make things easier for us to understand at certain later points.0584

All right, now that we understand what it is about, let's talk about axes.0589

The axes are just the vertical axis and the horizontal axis--those lines that we are graphing on.0593

The location of a graph can be as important as its shape.0602

The location is set up by its axes; we want to pay attention to these axes.0605

The axes will tell us where the graph is and what scale it has.0609

Often, our axes are going to be square; that is to say, the x-axis is the same length as the y-axis.0614

For example, we might have -10 to positive 10 on our y-axis, and -10 to positive 10 on our x-axis.0619

This is a pretty common one; and this is square, because the x-axis is the same length as the y-axis.0628

So, when we look at the picture, it is square, which is sort of an odd idea.0634

But if we made it so that they had different lengths, but we had set them out as the same amount of line,0638

then we would have a sort of squished picture; it wouldn't be the natural picture,0644

where we think of width and length as meaning the exact same thing, in terms of length.0648

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.0653

So, as long as it is square, the graph isn't distorted from the square perspective we normally expect.0660

However, sometimes it is going to be useful to graph functions on axes that are different from each other,0665

where we are going to want to have a really, really big y-axis, but very small x-axis--0671

where the function grows very, very, very quickly, so we want to be able to show all of its ability to grow.0679

But since it does it so fast, we need a short x-axis.0684

So, this is another really important reason to pay attention to the axes.0688

You want to know how long they are, what amount of information is being represented in both of them,0691

and also how big it is and where we are located.0697

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?0700

And just where are we located? Are we located in a weird place--is it not centered on 0?--those sorts of things.0706

So, let's look at a single function: let's look at f(x) = x + 1 and see how many different graphs0712

we can get out of it, just by changing the axes.0720

Just by playing with the axes, we can get totally different-looking graphs.0723

Here is the standard graph, our -10 to 10, -10 to 10.0726

This top left graph here is basically what our standard graph would be.0730

We are nice and square; the y-axis and the x-axis are the same length--that is what means to be square.0735

It is from -10 to 10 and -10 to 10--numbers that we are used to and expecting.0742

And also, the origin is in the center; we have (0,0) in the center of the graph.0747

Now, let's consider the one below that--the bottom left.0752

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 16.0756

-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,0769

it is still a square, because they have the same length, total.0776

-1 to 16 and -2 to 15 are both a length of 17; so it is still a square graph.0781

This one here is square; this one here is square, as well.0787

There is no distortion, no squishing in either the horizontal or in the vertical--no squishing of the graph.0795

And the origin, though, is in a totally different place than the center of the graph.0800

The origin is very, very bottom-left-corner; but it is still giving us the same x + 1.0805

It looks kind of different, in terms of where the axes are; but it is still pretty clearly the same function making the graph.0813

Let's look at another one: well, this one right here is actually not square.0819

Why? Well, we have totally different lengths here: -10 to 10 and -5 to 5--0824

that means the length of the horizontal is actually double the length of the vertical.0831

The origin is still in the middle, so that is nice; that is something we are used to.0836

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.0839

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.0847

This is not a square; these are not square axes right here.0855

Another one that is not square (and hopefully you can read the yellow)...0860

it is not too easy to read the yellow, but it is just me writing "not square" here.0864

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 20.0869

So, once again, we have not-square axes; but this time, we have the vertical axis being longer than the horizontal axis.0881

The horizontal axis has a length of 12; the vertical axis has a length of more than 20.0889

So, that means that we have stretched it in the horizontal; as opposed to being squished horizontally,0894

it has been stretched horizontally, because now it has less stuff to have horizontally than vertically.0900

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.0905

It has been squished vertically; so we have very different things here--vertical squish has happened in the bottom right one;0912

and in the top one, we have horizontal squish, but it is not because of anything that has happened to the function.0924

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.0933

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.0943

Unless we understand what the axes are telling us, we won't actually know what this picture means.0950

So, make sure you pay attention to axes; otherwise you can have no idea where you are.0955

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.0961

All right, one thing you might have noticed by this point is that the graphs in this course,0968

unlike this one to the right, do not have arrows on them.0972

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.0975

And that made sense; they were trying to get across a very specific point to you.0983

They were trying to remind you that the function keeps going on, even though we couldn't see it anymore.0987

In a way, we can think of the axes as sort of boxing in the function.0992

We don't get to see anything outside of the box of our axes.1000

But in reality, the function doesn't stop at 3; it doesn't stop at -3, necessarily; this is just a nice, normal parabola.1005

The function would keep going on; it would just continue off and off and off, and continue off and off and off and off.1012

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.1019

Just because the axes are here doesn't mean it stops; it is going to keep going.1026

So, that is what those arrows were for; at this point, though, I think you have probably gotten used to that idea.1031

We are not going to be using arrows at the ends of our graph in this course.1036

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.1040

We are going to assume that we are all aware that the graph keeps going.1047

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.1052

So, the graph is only stopping because the edge of the graphing axes stop.1061

It is the graphing axes that are stopping the function, not the function itself.1065

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.1071

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.1078

The only reason it stops is because it has hit this boundary at the edge of it.1087

It is not stopping because it actually stops; it is not stopping because the function stops.1093

It is just stopping because we are looking through a window.1097

If you look out through a window, if you are in a house, and you look out through the window,1100

you can't necessarily see everything to the left and everything to the right.1105

You can only see what you are currently looking through in the window.1108

You have to move how you are looking through the window, or move the location of the window1111

(although that would require a sledgehammer, and is something no one that you live with is going to be very happy about)--1115

you can move the location of the window and be able to see different things outside; but the window fixes what you can see.1121

That is what the graphing axes are doing to us: they are fixing what we can see in space.1126

We are not going to use arrows in this course, because we know that graphs have to keep going.1132

We are just seeing a tiny window on a much larger function.1136

That said, even though, in this course, we are not going to use arrows, and we are all aware of it at this point,1138

I want to point out that there are some teachers out there, and some books,1145

that will still use arrows, and will still require you to use arrows.1149

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--1153

doesn't mean that your teacher, if you are taking another course of the same type somewhere else--1162

that that teacher is going to be OK with it.1168

So, make sure that, if you have another teacher, if you have somebody else1170

who wants you to draw arrows--make sure you do what they are telling you to do.1173

So, do what they say as long as you are in their class.1176

For my class, you don't have to; we know what we are talking about.1179

But in somebody else's class, they might still want you to draw arrows, so be aware of that.1183

How do we actually graph? The easiest way to graph a function is by thinking in terms of that input-to-output.1189

Remember, you put in a number, and it gives out a number.1194

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.1198

For example, consider f(x) = x + 1, the one we keep working with.1203

If we plug in -2, that will give out -2 + 1, which is -1; so that gets us the point (-2,-1), right here.1207

If we plug in -1, that gets us 0; so that gets us the point (-1,0) right here.1215

If we plug in the point 0, then that gets us 1, 0 + 1, so that gets the point (0,1).1221

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).1226

And now we have a pretty clear idea: it is just a straight line; it is just going to keep going.1235

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.1239

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.1248

We don't have to graph all of the points perfectly in between, because it is pretty obvious,1255

at this point, that they would all just end up being on this graph, as well, if we were to keep going1258

with finer and finer steps, and how often we would check to see where inputs went to outputs.1263

However, straight lines are not necessarily the best way to connect all of our graphed points together.1270

In many ways, graphing is like playing a mathematical game of Connect the Dots.1275

But we don't necessarily want to connect with straight lines; we usually want to connect with curves.1281

For example, let's consider f(x) = x2.1286

Once again, here is a table that shows us input locations going to output locations, making points.1288

(-3,9), (-2,4),(-1,1), etc....we can see all of these points on this graph right now.1295

But let's look at what happens if we were to connect it all with straight lines.1301

If we connect with straight lines, we get this picture right here.1304

And while it is not a terrible representation of a parabola, it is not a very great representation of a parabola.1307

A real parabola has curves going on; it curves out; it curves out, as opposed to going out just in these straight, jagged lines.1314

So, we want to remember this fact: curves are normally what is going to connect our points, not straight lines.1323

The real f(x) = x2 is based on curves, so it looks like this picture right here.1329

It is based on these nice, smooth curves connecting all of these points together.1336

What about the fact that curves in one function are not necessarily going to look exactly like the curves in the next function?1341

That is true, but mostly, the graphs of functions are smooth; we want to connect points to each other through smooth curves.1346

So, whenever you are drawing a graph, make sure you are connecting things smoothly, without jagged, harsh connections.1352

Each function is going to curve in different ways.1358

Remember, the shape of a curve will be different: if we are using x2, x2 is going to give us1361

a totally different curve...well, not totally different, but it will be slightly different than x4,1365

which is going to be different than the cube root of x.1370

Each function that we graph will have a slightly different curve, or maybe a massively different curve.1373

But over time, you are going to become more familiar with the shapes of various functions.1379

As you graph more and more functions, as you see more and more functions,1383

you are going to think, "Oh, x2 should graph in this general way; √x should graph in this general way."1387

"The cube root of x, the x5...all of these things have curves that are slightly different."1394

It should curve a little faster, curve a little slower...those sorts of things.1400

Your previous experience with functions helps immensely, so just pay attention and think back:1404

when have I graphed something similar to what I am graphing right now?1408

And use that information to help you graph what you are working on at the moment.1412

Finally, the idea that more points make a more accurate graph: this is an important idea.1417

The more points you plot before drawing in your curves, the more accurate the graph becomes.1423

Each point on the graph is a piece of information.1428

So, it makes sense that, the more information we use to make our graph, the more accurate the graph is going to become.1431

Let's look at a specific example: Consider f(x) =...this complicated monster of a function, (x3 - 2x2 - 7x + 2)/x2 + 1.1439

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.1451

We are going from -4 to 4; so we will start at -4, and then we will step forward by 2.1458

That is what I mean by a step size of 2; don't worry--this is Δx; it means change in x,1464

and it is just a way of saying how much we are changing x each time.1470

So, if we step forward 2, if we go from -4 here to -2 here, and then to 0 here,1474

and then to 2 here, and then to 4 here, we have stepped forward by 2 each time.1483

And we can evaluate...I am not putting the table down here, because it is just kind of a pain1488

for us to have to see all of the numbers that we are going to be going through soon.1492

But if we evaluated each one of these things, we get the following vertical locations.1494

-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.1498

So, what happens if we increase the step size? We don't really have a very good idea of what this thing looks like.1508

It might go like this, but it could also go like this; it could maybe even do something crazy, like this.1512

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.1529

We are not used to this function, (x3 - 2x2 - 7x + 2)/(x2 + 1).1536

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.1541

So, since we don't have a really good sense of what it is going to look like, we don't have the expectations;1547

we need more points down before we are going to be able to have a good sense of where it is going.1551

Let's consider a smaller step size--a step size of 1.1555

Now, we go from -4 to -3, then -2, then -1, then 0, etc.1558

Now, we are starting to get a better idea of what the curve of the function looks like.1562

We are starting to think, "Well, now we are starting to see what is happening."1565

There is still a little confusion; we are not really quite sure what happens between -2 and 1 horizontal locations.1568

But we are starting to get a better idea; let's make it an even smaller step size.1574

We are at .5; oh, now it is starting to come in much clearer--we can start to understand what is going on.1578

We go with .2; oh, now we are really starting to see what it is.1584

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.1588

The only place where it isn't quite smooth is this section in the middle right here,1596

where the function is changing so quickly that we can actually still see the space between these tiny points.1601

But when it is not changing that fast, like most of it here or here, we end up seeing that it strings together,1606

because we have put down so many points that it basically turns into a smooth line.1614

And that is exactly what happens when we make a graph.1618

We are putting down so many points that we are saying, "Oh, that is what the smooth line is that it is making."1621

That is what is happening when you use a graphing calculator, actually.1626

If you use a graphing calculator, the computer inside is basically saying, "Make a bunch of points."1629

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.1635

So, it makes a whole bunch of points, and then it just strings them together; and that is what we see in the end.1643

The way that you graph something is: you just keep using more and more points if you need more information.1649

If you have a pretty good sense of how it is going to curve, though, you just have to put down enough points1653

so that you can then put in the curves, because you have already had the experience of working with that function before.1656

All right, when we introduce the idea of a function, we discussed an important quality for functions.1662

For a given input, a function cannot produce more than one output.1667

So, for example, we said that if f(7) = -11, then it can't also be true that f(7) = 20.1671

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.1679

So, if we put in f(7) the first time, and it gets -11, then the second time, it has to give -11,1686

and the third time it has to give -11, and the fourth time it has to give -11.1690

It can't ever be the case that all of a sudden, things go crazy and it produces a different result.1694

No, we can trust our function; we can trust our transformation, our process, our map, our machine--whatever analogy we want to use.1698

We can trust the function to always give us the same output if we put in the same input.1704

So, if f(7) = -11, it can't be the case that f(7) equals something else, as well--something different than -11.1710

We can turn this idea into a thing that we can see in graphs.1718

We call this idea the vertical line test, and it says that if a vertical line could intersect1721

more than one point on a graph, it cannot be the graph of a function.1727

So, if we have a vertical line, and we bring it along like this,1732

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.1739

No matter where we bring a vertical line down on this graph on the left, it ends up passing the vertical line test.1748

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:1756

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.1772

If we can do it at any place on the graph, even if there is only one place on the graph1780

where a vertical line hits the graph twice, then that means it is not a function.1784

If there is a vertical line that could intersect more than one point, it is not a function.1790

A vertical line--if it is able to intersect more than one location on the graph, it is not the graph of a function.1798

Why--why is this the case? Well, consider this.1808

Every point on a graph tells us where the x-value below is met.1811

The points on the graph are in the form (x,f(x)); the x that we put into the function, and the f(x),1814

the thing that the function puts out for that x--input and output put together.1821

So, for example, let's look at this graph: this is the graph of something like a square root function.1826

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.1831

So, we get that f(1) = 2, which is coming from the fact that the point is (1,2).1844

So, we put in an input, and we get the output of 2.1849

But let's consider this other one: what if we had this graph instead?1852

On this graph, at x = 1, we get (1,2) and (1,-2); that means, since it is a graph,1857

that if it is the graph of a function, we have f(1) = 2 and f(1) = -2.1866

But that is not possible--a function cannot give out two different things.1872

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.1877

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.1885

This picture right here is not the graph of a function.1893

Remember, the domain is the set of all inputs the function can accept.1898

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.1904

A graph shows where a function goes, so it means that we can see the domain in the graph.1914

Every point on the x-axis that the graph is above or below is in the domain.1921

So, every point on the x-axis that the graph is above or below has to be in the domain of that function.1926

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.1933

A really quick example: if we had √x like this, then if we have tried drawing a vertical line here,1939

that means that this horizontal location has to be in the domain, because it ends up having an output.1951

If we plug in this horizontal, it comes out as this output; so that means that it must be in the domain.1958

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.1963

That horizontal location is not included in the domain.1975

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.1978

Remember, the domain is everything that the function can take in.1984

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.1989

This is a great way to visually notice the domain; but we have to be careful to remember1999

that our function probably continues past the edge of our viewing window.2003

Remember the axes that we had there; so if we are going to use this idea,2007

we have to remember that, just because it seems to stop,2010

or we don't see anything past the edge of the axis, that doesn't mean that the domain stops there.2014

We just need to remember that it might continue on; we have to have some sense for how it looks beyond the edge.2018

We need to have some familiarity; we need to think, "Where would this keep going to?2024

Would this keep picking up those points in its domain, or would it stop for some reason?"2028

Range is the set of all possible outputs a function can have.2034

It is all the numbers that our function could possibly produce; so domain is what could go in; range is what can come out.2040

Like the domain, we can see the range of a function in its graph.2046

Every point on the y-axis that the graph is left or right of is in the range.2050

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.2054

So, for example, let's consider x2; x2 looks something like this.2060

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.2066

Similarly, we can go to this vertical location, and if we cut horizontally, there must be some domain location that puts that out.2074

Now, it turns out that there are actually two different domain locations that put that out; but that is OK.2083

Multiple domain locations--multiple inputs--can give the same output.2089

f(22) is equal to (-2)2; that is perfectly fine...4 and 4.2093

It is OK that the same input gives the same output; but the fact that there is some input that gives that output2101

means that it must be in the range, because it can be an output.2107

So, we go to any location on our vertical axis, and if we draw a horizontal line and it cuts the graph,2112

then that must mean that there is something that can input and give that output.2118

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.2124

If, on the other hand, we can draw a horizontal line on a vertical location, and it does not touch the graph--2137

that would not touch x2--then that means it is not in the range.2145

And that makes perfect sense: down here are the negative numbers.2149

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.2152

So, since there is no number that we can plug in to give out a negative number,2160

then that means that we can't output negative numbers, so they can't be in the range.2163

So, the range does not include any negative numbers, which is why, when we draw a horizontal line2166

in any of these negative numbers, it is not going to touch the graph,2171

because there is nothing that can make an output that would give a negative number.2174

Just like with the domain, we have to be careful to remember that our function probably continues past the edge of our viewing window.2179

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.2185

So, we have to have some feeling for how the function will look past the edges of what we are able to see.2193

Beyond the edge of our viewing window, we need to have some sense of what is going to keep going on.2198

If we have no idea, we need to expand our viewing window, so that we can have a better idea2204

and see, "Oh, yes, that would keep going," or "No, that actually stops."2208

Otherwise, we will not be able to figure out exactly where the range is.2212

Graphing calculators are really useful; if you haven't already noticed, this is a great time to point out2216

that there is an appendix to this course that is all about graphing calculators.2221

So, if you go the very bottom, and look at the appendix, there is an appendix about graphing calculators.2225

So, it might be at the end of the course, but that does not mean you should watch it last.2231

Graphing calculators are really, really useful for doing math.2235

And you can also use software for graphing on computers or tablets or phones.2239

And you can just start doing graphs on that really quickly and easily.2248

So, graphing calculators can be extremely helpful for getting a feel for how functions work.2251

If you are planning on taking calculus at some point, I definitely would recommend getting a graphing calculator in the near future.2256

You are almost certainly going to want a graphing calculator for calculus, and so it won't hurt to have it now in precalculus.2261

Even if you are not going to continue in math, you might find one useful for taking this course right now,2266

and maybe for other science courses that you are currently taking, or will take in the future.2272

So, if you are interested in getting a graphing calculator (and I would recommend it if you can afford it--2276

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)--2280

check out the appendix on graphing calculators; we are going to talk about all about2286

how you can use them, what they are good for, why you might want one,2290

what are some recommendations, things to look for, and that sort of thing.2295

So, check out the appendix; there is a whole lot of information on graphing calculators there.2298

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,2302

since you are taking this course, and there is a very good chance you will go on to take calculus.2306

I would definitely recommend to get a graphing calculator if you can afford it.2310

So, check out the appendix; there is lots of information there.2314

All right, we are ready for our examples: first, we are going to graph something.2318

Graph f(x) = x2 - 3x + 1: we have done this before, but let's just see a quick reminder.2322

We want to do this by plugging in points and getting outputs.2327

So, we are going to plug in x's, and we will get f(x)'s out.2332

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.2335

If we plug in 0, we get 02 - 3(0) + 1; that gets us 1.2344

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 -1.2356

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 + 1.2373

4 + 1 is 5; 5 - 6...we have -1, once again.2385

Let's try going in the other direction as well: let's plug in -1.2390

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,2394

or be able to do it on paper on your own, I'm sure; so we will just speed things up.2398

(-1)2 - 3(-1) + 1...that will get us positive 1.2403

We plug in -2: (-2)2 - 3(-2) + 1; (-2)2 gets us 4, minus 3(-2) gets us...2409

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.2425

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."2430

(-1)2 gets us positive 1; minus 3(-1) gets us positive 3; plus 1 gets us 5.2441

(-2)2 - 3(-2) + 1...we have 4 + 6 + 1; we have 11.2451

And if we go forward one more, at 3, we are going to see 32 - 3(3) + 1.2464

We would get 9 - 9 + 1, so we would get positive 1.2474

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 1.2479

So, 16 - 12 is 4; with 4 plus 1, we get 5.2490

All right, so we have a lot of information, but there is one thing that we might notice.2494

We might say, "Parabolas need a bottom"; we are graphing a quadratic, and while we haven't formally talked about them,2497

I am sure you have seen parabolas quite a few times by now.2505

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.2509

That is going to give us sort of a flat bottom, so there is probably some point in between them that is even lower.2516

So, we want to have some sense of where it is going; so let's actually plug in a number in between them.2522

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 -1.25.2526

I will actually do it here; so we plug in 1.5, so 1.52 - 3(1.5) + 1...2540

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 + 1.2548

So, we have 3.25 - 4.5; we get -1.25; great.2557

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.2563

But we don't know that technically; we haven't formally talked about it.2569

But at least it gives us a sense of where this is going to be bottoming out.2571

So now, let's actually set up our axes, and let's plot the thing.2575

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.2578

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.2588

But we are going to top out, so we will never actually end up seeing the number 11,2603

because we can't make it up that high on these axes, if we are going to keep them at this reasonable size.2606

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.2610

So, this is approximately square; I am just roughly drawing it by hand, but it is pretty good.2617

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,2621

because -2 is already out of where we are going to be able to plot.2635

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.2639

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.2649

We plug in 1, and we are going to be at -1; we plug in 2, and we are going to be at -1.2658

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.2665

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.2673

So, we have some sense of how the curve looks.2680

All right, and it would keep going on and out; and it just keeps going, past the edge of our axis.2686

All right, and that is how we graph it; so this is pretty much how we can graph anything.2698

Plot some points on a T-table; plot some points on x and f(x), input and output.2702

Plot the points; figure out where they are going to go; then actually put them onto the graph.2708

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.2712

All right, this is the graph of f(x) = x3 + x2 - 6x.2720

Using it, we are going to estimate the values of f at -1.8, f at -1, f at 1, and f at 2.5.2725

Then, we are going to also estimate the values where f(x) = 0; and then finally, we will estimate the values where f(x) = -3.2732

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.2738

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 9.2748

If that is the case, I would say that looks like around 8.3 to me, give or take.2763

We are just estimating, so we don't have to be absolutely, perfectly precise.2767

But I would say 8.5 is a pretty reasonable guess; 8.3 is probably a little closer, so let's go with 8.3.2769

f(-1.8) is equal to...it looks like 8.3; it is an estimate--it says "estimate"; it is a graph--2776

we are never going to be able to perfectly pull information from the graph.2784

Well, we might be able to in a few cases; but it is going to be normally something2787

where we are getting that we are pretty confident, but it might be slightly off by .01 or .01...2790

well, that is the same thing...by .1, lower by .1, these sorts of things.2796

It is hard to be absolutely, perfectly precise, since we are looking at a picture; but we can get a pretty good idea.2800

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.2804

So, I would say probably about 5.7 or 5.6; so let's say it is 5.7.2814

f at 1...we are here; we drop down; and that one looks like it is really pretty much exactly -4; so f(1) = -4.2824

f(2.5)...plug in 2.5; we go up pretty high; that looks like it was a pretty good vertical...2836

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.2849

We have estimated the values for all of them; they might be a little bit inaccurate, but they are pretty close to right.2858

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.2863

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.2867

Now, let's look at estimating the x-values for f(x) = 0; we will do this one in blue.2876

Estimate the x-value for f(x) = 0; so what is f(x) = 0?2882

Well, remember, the vertical axis is f(x); that is the output.2885

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.2889

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.2896

There is nothing else crossing there; so we can assume that we have found all of the x-values.2905

It seems that is going to be x = -3, 0, and 2; they all caused f(x) to come out as 0; great.2909

Finally, we will use red for the very last one; hopefully, that won't be too confusing.2924

Estimate the x-values for f(x) = -3; if that is the case, we go to where f(x) is -3.2928

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.2934

And there is something; so f(x) = -3 at these three horizontal locations.2955

So, once again, it is not absolutely, precisely, absolutely perfect, but pretty good.2965

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 -3.2.2972

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 0.5.2982

That is right here, so I would say that is just a hair past 0.5, so let's say that is 0.6.2998

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 1.65.3004

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.3016

That is how we use a graph to figure out things from it.3025

We can estimate values given an input, or we can estimate values given an output.3027

We figure out what makes that output or where that input would get mapped to.3032

What would that input get output as?3038

Vertical line test: Which of the below is not the graph of a function?3042

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.3046

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.3053

The vertical line is never going to cross it at anything.3060

The only place where you might be a little curious is right here where it curves up.3062

But it never really continues on in such a way that we can be sure.3065

Any vertical line that we are making seems to cut it just once.3069

Now, it does have this part where it sort of curves like this, but that is inaccurate.3073

It looks like that, but the graph is actually curving a little more like this.3078

And the reason why it looks like it is stacked on top of itself is because we have to add thickness to our line.3082

In reality, the line is actually thinner than that; and is even thinner than that, because a point is infinitely thin.3088

So, there is no stacking, because of that infinite thinness; it is only because of that thickness of our line3094

that it ends up looking like there is something stacked.3099

So in reality, if we come along with the vertical line test, since the vertical line is also infinitely thin,3101

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.3106

So, this is a function.3112

If we cut in the middle, it is going to hit a bunch of times.3121

It cuts here, here, and here; so that is more than 1 intersection.3127

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.3131

So, in the middle, it manages to fail being a function, because one input manages to simultaneously have three outputs.3139

So, it is not a function.3148

Finally, this one over here is the same idea as the left side.3154

Even though it looks like it is starting to get vertical, it is never actually vertical at any point.3159

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.3164

So, we have to think about this, beyond just saying, "Well, it looks kind of stacked, so it must be."3171

No, we have to think, "Oh, that is really only approximating where the graph is, because the line,3175

while we can't see infinitely thin things...that is what the line is representing."3180

So, it is the case that this one is also a function, because there is nothing where it clearly cuts two places at once.3185

Great; that is how we use the vertical line test.3191

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.3193

If we can drop vertical lines everywhere, and it would never cut the function more than once, then it is a function.3201

Final example: Prove that there is no function that could produce a circle as its graph.3209

This might seem a little complicated at first; so what we want to do is think, "Well, how could we prove this?"3213

Well, if we want to prove it, we need to show something involving circles as graphs.3218

We get stuck too much on trying to think, "What is the right way to do this?"3224

We might never get any progress.3227

But if we think, "Well, what does a circle look like?" a circle has things stacked--it would fail the vertical line test.3228

So, we know we can prove this by contradiction.3237

Proof by contradiction: we are going to start by assuming that there is such a thing.3241

So, proof by contradiction: assume such a function exists.3249

If there is a function that could produce a circle, then look at its graph.3257

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.3268

So, here is a circle; and while it is not a perfect circle--I am but a mortal--it is a good idea.3278

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."3284

The graph must fail the vertical line test; therefore, it is not a function--it cannot be a function.3294

The graph cannot be a function; but it was the graph of a function.3314

So, since the graph cannot be a function, it must be that no such function exists.3322

So, our assumption was that the function did exist; since the graph cannot be a function,3335

but it was just the graph of a function, then there is a contradiction.3340

The function cannot exist, so it must be that no such function exists; and we are done--that is our proof.3343

All right, assume that what we can see on the graph below is the entirety of the function f.3350

In other words, there is nothing past the edge of the axes.3355

We are looking through that window, but we have been told that there is nothing interesting past the edges of the window.3358

So, this graph here is the entirety of the function f.3363

Now, we want to estimate the domain and range of f from the graph.3367

Now remember, the domain was everything that can be input.3370

So, if we go to, say, 0, look: 0 shows up in the graph.3374

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.3381

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;3390

but it looks, probably, like 3.5 gets put in, so we would say that the domain is going to go from -2 to 3.5.3399

What about the range? Range is everything that can be output.3410

Is there anything that can output at 1? Yes, 1 manages to touch here, and manages to touch here.3416

There is some input that gives out 1; if we put in an input here, we can see that it connects here.3426

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 3.3433

So, 3 is not in the range; the highest that we manage to get to is right here.3442

So, it looks like 1.5 is the highest that we managed to get to with the graph.3448

It never shows up over here, but that is OK, because it shows up somewhere.3453

And then finally, it looks like the lowest we manage to get to is -2.3457

So, our range: the lowest location on our range is -2, and the highest location that we manage to make it to is 1.5.3462

And we hit everything in between: if you go to any higher location in between, it shows up.3471

So, our range is everything in between -2 and 1.5, because all of them have something that they are able to contact; great.3477

All right, I hope you have understood what is going on here; I hope it has really crystallized the idea of a graph.3484

Graphs are so important; they are going to show up in so many things in math.3489

And they are also going to show up in science, and even if you just look in a newspaper.3493

Graphs make up a really, really big part of mathematics,3496

so it is really important that we understand what is going on with them now,3499

because we are going to see a lot of them as we go on.3502

All right, we will see you at Educator.com later--goodbye!3503