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For more information, please see full course syllabus of Pre Calculus
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Lecture Comments (1)

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Post by Jim Hayes on October 11, 2017

You do know that they didn't edit out your mess up at the beginning right? It was funny.

Graphing Functions, Window Settings, & Table of Values

  • For the most part, graphing functions (or equations in the form y=) is pretty direct. Just enter the function into the graphing calculator and graph it.
  • The graphing calculator usually expects the variable to be x, so if your function has a different variable, just change it over when entering the function:
    f(t)  = t2 + 5t −3     ⇒     f(x)  = x2 + 5x −3
  • Just like calculating values, the syntax we choose will affect how the function graphs. Be careful about syntax and use parentheses as necessary to ensure that the calculator does what you want.
  • One of the most important ideas when graphing is to think about the viewing window. Since a graph goes on forever, you can only ever see a portion of it, which is what you have in your viewing window. Most graphing calculators will start the viewing window looking at x: [−10, 10] and y: [−10, 10] or something similar. But often that's not a very useful way to look at whatever function we're working with. You can adjust the viewing window by zooming and scrolling, but often it's faster and easier to just manually set the window. Go to window settings, and put in the values that will allow you to see your function best.
  • One great use for a graphing calculator is the table of values. This allows you to quickly churn out values for various inputs on some function. After entering a function, you can hit the table button to have it display a table of values.
  • Often you want to have more control over what numbers the function uses for inputs. In that case, go to the table settings and change the independent variable (x) from `automatic' to `ask'. With it set on `ask', the table starts off blank. You can enter whatever value you want for the input, and it will give you the appropriate output.

Graphing Functions, Window Settings, & Table of Values

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
  • Graphing Functions 0:18
    • Graphing Calculator Expects the Variable to Be x
  • Syntax 0:58
    • The Syntax We Choose Will Affect How the Function Graphs
    • Use Parentheses
  • The Viewing Window 2:00
    • One of the Most Important Ideas When Graphing Is To Think About The Viewing Window
    • For Example
    • The Viewing Window, cont.
  • Window Settings 3:24
    • Manually Choose Window Settings
    • x Min
    • x Max
    • y Min
    • y Max
    • Changing the x Scale or y Scale
    • Window Settings, cont.
  • Table of Values 7:38
    • Allows You to Quickly Churn Out Values for Various Inputs
    • For example
    • Changing the Independent Variable From 'Automatic' to 'Ask'

Transcription: Graphing Functions, Window Settings, & Table of Values

Hi--welcome back to

Today, we are going to talk about graphing functions, window settings, and table of values.0012

For the most part, graphing functions or equations that are in the form y = stuff involving some x is pretty direct.0017

Just enter the function into the graphing calculator, and graph it.0025

The graphing calculator expects the variable to be x; so if your function has a different variable, just change it over when entering the function.0029

For example, if you have something like f(t) = t2 + 5t - 3, when you enter it in,0035

you should enter in x2 + 5x - 3; you just swap out all of whatever the variable used to be for x,0040

because it expects it to be x when you plug it in.0047

But this has no effect on what shape the graph is, so there are no worries there.0049

Just like calculating values, the syntax we choose will affect how the function graphs.0054

If we wanted to graph the function f(x) = 22x, we might be tempted to enter 2, exponent, 2x.0058

Remember: this right here, the carat, means "exponent"; if we have 37, that is saying "three to the seventh."0066

So, 22x...we think that that will be 2 raised to the 2x; however, that would actually give us 22, times x.0073

It sees that as 22, and then the next thing entirely.0082

Instead, to get the correct entry, we have to use parentheses to indicate how it is all put together.0086

So, we wrap our 2x in parentheses; so it sees that we are raising 2 to the entire thing of 2x.0091

2x together is our exponent; and the parentheses show that it is that whole object that we are using as our exponent.0098

So, it is really important to use parentheses when you are not quite sure how the interpretation will work out.0105

Once again, better safe than sorry; and extra parentheses are not going to hurt you.0109

But if you miss parentheses in an important place, it can really mess things up.0113

The viewing window: one of the most important ideas when graphing is to think about the viewing window.0117

Now, since a graph goes on forever--it never stops going on--you can only ever see a portion of it, which is what you have in your viewing window.0121

Your viewing window is the portion of the graph that you are seeing.0129

Most graphing calculators will start the viewing window looking at x going from -10 to 10, and y going from -10 to 10, or something close to that.0132

But often, that is not a very useful way to look at whatever function we are working with.0141

It works well for a lot of functions, but it is not going to work for every single function.0145

For example, consider if we wanted to graph the function f(x) = √(x - 17).0148

Well, if we graphed that function on normal axes, those right here (sometimes called "standard zoom"), we would see this.0153

It doesn't show up on the viewing window at all; f(x) = √(x - 17) with x going from -10 to 10 and y going from -10 to 100162

doesn't show up at all in a standard zoom, in our standard, normal way of looking at it,0171

because √(x - 17) isn't defined for any x-value going from -10 to 10.0176

So, it doesn't end up working; so, just because they are the standard axes,0182

and just because those are probably what we are going to get the first time we graph with something,0186

doesn't mean that we should assume that that is the only part of the graph.0189

There are other parts to go on; and depending on a specific function, the normal, standard part might not be the interesting part at all.0192

While the graphing calculator will allow you to zoom out (if you get something where you don't really see the whole thing,0199

you can always press the zoom button and zoom out until you can see the part that you are looking for,0203

and then you can zoom back in on the section you care about, or put a box so that it zooms in just on that specific portion),0208

that can be a slow, cumbersome process, especially depending on the speed of your calculator.0213

Some calculators take a while to do a zoom; and so, it could take you 30 seconds or 60 seconds, just to get to the section you actually care about looking at.0217

Similarly, depending on the calculator, you might be able to scroll the window over--be able to click it over,0226

one piece at a time, and sort of nudge it in one direction or another.0231

But once again, this process can end up being pretty slow.0234

If you have a not-very-powerful graphing calculator, it has to re-draw it every time you nudge it over.0236

And if it takes a little while for it to draw out the graph, that can, once again, be thirty seconds or a minute0241

of just waiting for it to go, step-by-step, over and over.0245

Plus, you might not even be sure which way the interesting stuff is going to happen, if you just immediately try to scroll it around.0248

Instead, I say (at least--other teachers might say other things, but personally, I recommend):0254

the best option is usually to just manually choose your window settings.0259

Go to the window settings, and take direct control of the viewing window. Tell it where you want to look.0264

If you look at the window settings, the important parts are these four: xmin, xmax, ymin, ymax:0271

where your horizontal axis starts; where the horizontal axis ends; where the vertical axis starts; and where the vertical axis ends.0278

By changing the values given for these, it will move the placement and size of your viewing window accordingly.0286

This gives you direct control over what you are looking at.0290

Also, because you have direct control, when you look at the graph, you will already know how you framed this:0293

how big is it horizontally? How big is it vertically?0299

So, you can have a sense for what the aspect ratio will be like.0301

You might also be interested in changing either xscale or yscale, the length that tick marks are spread on their respective axes.0305

By "tick marks," I mean those little ticks that say how many units we have moved.0312

Usually, you don't need to actually care about the tick marks.0317

But sometimes, they can help for seeing some types of function.0319

It might be handy to have a sense for just how far you are out.0322

For example, in trigonometry, it often helps to set xscale equal to π/2,0325

because very often, the interesting stuff happens at π/2 or something along those lines of π's scale.0330

So, it can be handy to set a specific tick mark value, but not necessary.0336

By thinking about the function we are graphing, we can usually figure out a good starting place for the window.0342

So, going back to our example of f(x) = √(x - 17), we know that it won't start before x = 17,0347

because if we put in x = 17...less than that, if we put in x < 17,0354

it is going to be undefined, because we will have negatives in our square root.0360

So, we know that it won't start before x = 17.0363

We also know that it only outputs positive numbers, because it is a square root;0365

and so, square root there is only going to be able to allow us to get positive numbers out of it.0369

And finally, we know that it is going to grow very slowly, because we have probably graphed a square root function before.0373

And we know that they slow down, the farther we get out.0377

Taking all of this into account, we might set our graph of f(x) = √(x - 17) with our x-value going from 0 to 40,0380

because we know that it is going to be a while before anything interesting happens.0388

We have to go at least past 17 to see anything show up on the graph at all.0391

And since it grows slowly, we will probably want to go out to a fairly large x-value.0395

We can set our y from 0 to 5, because we know, since it grows very slowly,0400

and since it is only positive, that there is not going to be a whole lot of action happening vertically.0404

So, we might as well just set it to a fairly small thing, so we can see exactly what is going on there.0409

So, we graph it like that; and we would get this graph right here.0415

However, notice: just because the graph looks square doesn't mean that the aspect ratio actually is square.0418

In this case, our y manages to go only from 0 to 5, while our x manages to go from 0 to 40.0424

So, for this specific picture that we are seeing right here, we see that it has been sort of squished horizontally,0431

because it has a short distance vertically, so it seems like it is moving a lot more vertically.0437

But in reality, it is actually growing out very slowly.0441

So, by paying attention to the window settings as we do them manually, we will have a sense0444

for what the picture we are looking at actually means, in addition to being able to quickly and easily control where we will be looking.0448

A great use for a graphing calculator is the table of values.0455

The table of values allows you to quickly turn out values for various inputs on some function.0459

For example, if we have f(x) = 4x2 - 7x + 9, we could hit the Table button.0463

We go to look at the table, and it would display something like what we have to the right, where we have x and f(x).0469

So, we would see that -3 gets a 66; -2 gets us 39; -1 gets us 20; 0 gets us 9; etc., etc.0475

If we want additional values, we can just scroll up or down the list to find more.0483

If you take your cursor, and you push one way or the other, it will normally be able to show you more of the values one way or the other.0487

It will depend on the specific calculator you are using and how it works.0493

If it doesn't quite end up working for you, do a quick Internet search, and you will be able to figure out how to get other values on your table.0496

However, the most useful thing for a table of values is what I am about to tell you.0501

Of course, often, you want to have more control over what numbers the function uses for inputs.0505

Having it go in only integer steps is sometimes useful; but very often, you want to actually tell it what numbers to map.0511

And if it is an integer input, you can probably do it by hand pretty easily.0517

What you care about is the difficult numbers to plug in, like decimal things.0521

In that case, what you want to do is go to the table settings: find the table settings on your calculator.0525

And go the independent variable; change your independent variable, which is just the x--what you are plugging into the function.0530

And change it from Automatic, what it normally starts at, to Ask.0537

You are changing it from it setting it for you to it waiting for you to tell it what value to use.0541

With it set on Ask, the table will start blank; you can then enter whatever value you want for the input.0547

And then, it will give you the outputs from the inputs that you give it in.0554

For example, if we have f(x) = 4x2 - 7x + 9, we might be curious, for some reason, for the problem that we are working on:0558

we want to know what 4.7 is; so we put it in--we type it into our calculator: 4.7 for the input.0564

We hit Enter, and it says 64.46.0571

We could enter whatever we end up wanting to know.0575

We might be interested in 12, -3.9, π...we can just plug in the various numbers.0577

We hit Enter, and it just immediately cranks out what comes out of the function.0582

This setting makes it extremely easy to just churn out massive numbers of values for a table.0586

Making tables comes up a lot in math, and this makes it so much less tedious.0591

Being able to just set up our function once, and then just put a number, put a number, put a number, put a number...0597

and then we can just write the things down in a table of values.0602

If we tried to do it each time by hand, we would have to do 4 times 4.72 minus 7 times 4.7 plus 9...0605

Even if we are using a calculator, but we are working through calculations, it is going to take a long time to do one of these after another after another.0612

It is normally so much easier to just set up a function that does what you want to do in general,0618

and then just put in the various values that you are interested in looking at.0623

That way, you can just churn out all of the values really, really quickly, and it doesn't take a long time.0626

It is easy; it is much more comfortable that way.0630

All right, in the next lesson, we will look at how we can find some of the more interesting points on our graph.0632

All right, we will see you at later--goodbye!0636