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Robert McKeown: The slides have some tips on how

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to transform graphs. Essentially, you want to

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think about having a function in its general

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form like that. And then there's two

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manipulations that might be made. One is to

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double the function itself. In English, what

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does that mean? Well, if you have the same

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amount of f of x, right, if x is the same value,

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you're going to get twice as much y. Now, what

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happens if we put that now notice that the two

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here is outside the function?

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Looking at the next example, we're inside, it's

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inside the function. Well, what does that mean?

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Well, that means that you can have the same f of

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x the same y with half as much x half as much x.

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So we're trying to do that to give you a little

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bit of intuition. So here we are on ALEKS, we've

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got a question. And it says that we've got one

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half of X the function of one half x, this means

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that we're going to need twice as much X to get

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the same amount of y. So if I want to get y

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equal to negative four, and before, that would

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require two X, now it's going to take four x, so

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I'm going to draw a little blue cross make a

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little point there. Now, if we can see by the

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original function here, that if x is equal to

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zero, y will be equal to zero. So multiplying x,

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or sorry, zero by one half is not going to

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change that. So the origin stays the same. And

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similarly, on the other side of the diagram, if

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negative two x got us negative four y, now it's

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going to take negative four X to get negative

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four y is going to take twice as much X to get

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us there. I'm going to just draw the lines, now

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I got to draw two lines. And so we stretch the

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function out to the left and the right, because

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we need more X to get the same y. So let's see

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if that was correct. And it was correct. So we

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needed twice as much X to get the same amount of

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y. So a parabola looks something like this.

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Or maybe like this. And it just keeps going off

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in that direction. This form here would be

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something like y is equal to negative two x

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squared plus four x plus four, something like

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that. And when this is negative, we're going to

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get out max up here. The one on the right, could

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be something like two x squared minus four x.

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And it looks like maybe negative four. Something

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like that. And where if this term is positive,

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this is gonna be a min and so this one over here

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is decreasing. As the absolute value of x

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increases, and over here, y is increasing as the

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absolute value of x increases. Here, the max it,

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this is the vertex. The vertex is a max here.

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The men is also the vertex, the vertex is the

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min. So the question is asking us to find the

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vertex. There are a few ways to find the vertex.

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I'm just going to show you how to do it by trial

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and error. Essentially. There are other ways but

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we don't use them. They don't come up very much

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in economics, so I won't spend too much time on

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that unless you really want me to. You're more

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than welcome to ask me questions. I guess via

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email if we're doing distance or after class if

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we're in person. So for trial and error, I

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usually just, I will just guess that x is equal

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to zero. If x is equal to zero, y in this case

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is going to be equal to negative two. parabolas

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have certain properties that are going to help

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me with my gas, I know that this parabola, the

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square term is positive. So the vertex is going

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to occur at the minimum, so I can plug and play

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with some numbers and see if I can find that

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minimum. I know that a parabola has a certain

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amount of symmetry, if I draw a straight line

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coming up from the vertex, then the distance

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from that straight line to each of its wings is

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going to be the same. Looking back at my guess,

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when x was equal to zero, y was equal to

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negative two. Well, why don't I guess? Okay,

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well, what happens if x is equal to one, well,

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then I get three times one squared, plus six

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minus two, and that's going to give me seven. So

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it's quite a bit higher than negative two. Now,

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if I can show that x is at negative one is

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greater than y is equal to negative two, then

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I'll know that at x zero, y equal negative two,

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we've got the vertex. So I'm going to guess x is

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equal to negative one, I get three times

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negative one squared, plus six times negative

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one, minus two, and I end up with minus five.

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Now I know that x zero, y negative two is not,

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well, maybe I should make it a little more

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formal here. I know that the pair's zero,

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negative two is not the vertex.

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But all hope is not lost. Because the coordinate

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negative one, negative five, might be the

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vertex.

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So I'm gonna guess. x is equal to negative two.

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And I've got y is going to be equal to negative

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two squared times three, plus six times negative

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two, minus two. And I've got 12 minus 12 minus

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two, which equals to negative two. So therefore,

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negative one, negative five is the vertex. I did

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that by guessing. And I know that Cindy has a

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problem with the square term, this term right

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here being positive. I know that once I find the

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minimum point, that's going to be the vertex.

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And because of symmetry, any other value of x is

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going to give a higher value of y. We can now go

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over to ALEKS. And I'll show you how to graph

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this parabola and ALEKS, anywhere feature and

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ALEKS. And that's that this little diagram here

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with the X on it, so I'm going to click on that,

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and it says, plot a point anywhere. And I'm

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going to go down to here. And I'm just going to

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put in all our coordinates

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that we calculated before. And notice that every

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time I I do that and I click the plot a point

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button, a little Little x appears.

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Okay, so I've plotted the points of the

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parabola. And then I'm good. All I had to do is

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press this button right here. And ALEKS has sort

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of knows what to do with the plot points. Now

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let's see if it's correct. And it was correct.

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So we got the right answer.