Transcript
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The speaker reviews the function standards already covered in the unit and identifies the remaining topics to address. The lesson transitions into understanding patterns of change in functions.
Okay.Yay.Okay.
So, as we're going and looking through the standards for this unit on quadratic equations and and functions, we've done quite a bit.Uh we've done polynomials, exponentials, logarithmic.Uh I would say done absolute value.We haven't done rational.We haven't done trigonometric.And we haven't um talked about discrete versus continuous uh domains.
So, we'll get um I it's been implied, but I will make it a little more explicit when we do this so that's better.Get it out of the way.Come on, move.Doesn't want to.So, let's go on to the next one.We've done tables, graphs, equations, descriptions of We've done recursive.We've talked about finite differences.Uh we've talked about zeros, intervals of increase and decrease.Do you guys remember this?Okay.Uh average rate of change.Yeah, we've done that.Um that's just a difference in the Y's over the difference in the X's.It's actually the slope if you're linear.Um We've talked about domain, range, and end behavior.Um and we've talked about shifting, the transformations, remember?Also compressing and expanding or um what do we call it?Shrink.I would say a shrinking thing is not usually a shrink.So, we've talked about all of those transformations.
Um but what we haven't talked about is inversely proportional and directly proportional.And so, we're going to kind of go to that today and I'm going to start off with these notes.And then we'll kind of go through with the the rest of the standards as we go along, but find this notes
This section reviews the defining rules for polynomials and then introduces inversely proportional relationships. It explains how inverse variation works and shows the graph of y = 1/x as a hyperbola.
understanding patterns of change in functions.So, one of the things I I wanted to remind you of are the core rules for polynomials.So, they look like this.f of x equals um a sub n x to the n plus a sub n minus one x to the n minus one plus and it just keeps on going till you get to a zero where the youdon't have any x's left and you just have a constant.Um we have too many dots.Remember in mathematics, if you have more than three dots, you just being overly exuberant.So, we don't want to do that.Now, what are some of the conditions for polynomials?Well, the exponents have to be whole numbers.So, I cannot have an exponent, if it's a polynomial, that's 1/2.None of the exponents are square roots or anything like that.They have to have whole number powers.
So, that would be 0 1 2 3 and on.They can't have non-negative integers.I mean, they have to have non-negative integers.They can't negative integers.
And you can't have variables in the denominator.So, you can't have like f of x equals 3 um times 2 over x plus 4 times 5 over x squared.You can't have stuff like that.This is a no.Right?You can't have variables that are in the denominator.And so, that's another uh throwback to here.You can't have negative exponents.This would be x to the negative first, x to the negative second.You can't have negative exponents.Cuz that puts the x in the denominator.You can't have fractional exponents cuz that would be like square root of x.So, that wouldn't work.It's not a polynomial.It is something, but it's not a polynomial.And you have to have real number coefficients.So, you can't have 3i.Right?No imaginary numbers.You can't have You can't have that.So, your coefficients have to be integers, fractions, or decimals.
You could have transcendental, but we usually don't.Uh the expression must have a counted ending number of terms.You can't have an infinite number of terms.It's not a polynomial anymore.It's a series.So, if it has an infinite number of terms, you've got a series.Not a polynomial.And then, inversely proportional relationships.I bet you already know this.So, if it's inversely proportional, we're saying as one thing goes up, another thing goes down.Like as my taxes increase, my amount of spending money goes down.Those are inversely proportional.Okay?What's another Can you think of another example?
I guess I do more weight lifting, my personal weight seems seems seems to go down.Now, after a certain point and you're building a lot of muscle, it might not be that relationship, but just you know, trying to tone up, you might find that that is inversely proportional.So, what happens is when you multiply X and Y, you're going to get a constant all the time.That's inversely proportional.So, Y equals K over X if you divide both sides by X.You you usually see it like this.K is called a constant of proportionality.
So, here's an example.The time taken to complete a task decreases with the increase in the number of workers finishing it if they're all working on it.And would decrease and if you had well, it the time would increase the fewer workers you had, right?It's always opposite.
Now, what happens when we draw these things?What is the graph of Y equals 1 over X?It's this hyperbola.That's the hyperbola.Can you see it?Thank you.Yeah, Y equals 1 over X, that's a hyperbola.
And then if I put in a negative value, it will reflect it.So, these just reflect down.This reflects down.This reflects across the Y axis.I started across the X axis, this axis.
The speaker explains the parameters of exponential and logarithmic functions, including growth, decay, and domain restrictions. The section also shows how exponential and logarithmic graphs are inverses reflected across y = x.
So, now let's talk about the parameters of an exponential function.We talked about this before.If you have an exponential function where it's a times b to the x, the a is the vertical stretch or if it's a fraction, a vertical compression.If it's, you know, between 0 and 1, it's going to be compression.And then if it's negative, it'll reflect it, of course, across the x-axis.The y's are flipped down to the opposite part.Um So, that's what happens with the a.What about the b?Well, b controls growth or decay.So, if b is less than 0, we have decay, or I mean less than 1.If b is less than 1, but it has to be bigger than 0, that's decay.Um Um that's what it was said right there.
And then what about x?What about x?X has to be what?Can I put negatives into a log?No.Can I put 0?So, x has to be greater than 0.Can't put negatives, you can't put logs.I don't know what those negatives are.And 0.So, then what are the parameters for a logarithmic function?Well, if you have a logarithmic function, um it's the inverse of an exponential function.So, they're a reflection across the y = x line.So, for example, if I have this log, the a control controls the vertical stretch or compression.
So, if it's If it's less than zero, what will happen?
What do you think?Wouldn't it just be a reflection?Yeah, if it's negative, it's just a reflection.
How about the base of a logarithm?Well, it has to be greater than zero.It can't be one cuz if the base is one, then any number you put for the exponent, the answer is going to be one.Um and then we had the common log base log and the natural log base ln.So, you guys remember that.Here's just some key features for the exponential function, the domain is from minus infinity to infinity and the range is going to be from uh zero to infinity.
You're not going to have any negatives.How about this?What about this?Well, it has a um horizontal asymptote y equals zero and the inverse has an asymptote at x equals zero for the logarithmic function.They're almost the opposites, aren't they?Aren't these the opposites?See how it's exchanging the domain and range when you do the inverse?That's what the inverse is.You've exchanged the domain and range.So, a key point on this um exponential function is zero one.A key point on the logarithmic is one zero.Everything's just the opposite, right?And if I have log b um and inside is b, you're just going to get x.Remember these will undo each other.And these also undo each other and you just get x around that.Okay.So, let's talk about in a particular example and I didn't write this, so you'll have to do it.
So, let's graph f of x equals 3 * 2 to the x and its inverse and then graph x equals y y equals x as a dashed line.I I didn't even follow my own yet.own directions.This needs a dashed line.And I didn't dash it.Uh so, how do you find the inverse of a function?Well, if this is the function, we we swap the y, this will be my y.We'll swap y and x.So, where I see x, I'll put y and where I had f of x, I'll put x.So, I just swap x and y to find inverses and then I resolve for x.Now, I solve for x.So, the first thing I want to do is maybe take the log base 2 cuz that's the base here.Take the log base 2 both sides.And so, this will be log base 2 of x over 3 and then remember these two undo each other and I just get y.
So, now I have my inverse function.This is y equals log base 2 x over 3.That's my inverse function.And then I just put some points in like I put in 0.So, if you put in 0 here, you're going to get 2 to the 0 is 1, so 1 * 3 is 3.So, if I put in 0, I get 3.And this does tell me the y-intercept when I have an exponential function.And then um I think I put in um What did I put in?I think I put in 1.If you put in 1, you get 3 * 2 is 6.So, I put in 6 and I said, "Okay, I've got enough to I know what it looks like, right?"I just drew it with those two points.That's enough for me.And then I looked at the inverse function and I said, "Ooh, I don't really want to have to work too hard.
What goes into What does three divide?Six is something three divides.It goes in there twice, right?So, two to the first power would be two.So, if I put in six for X, what power out here would give me two?Two to the first, right?So, one.So, six one, I graph that.Six one is here.And then I did 12.So, three goes into 12 four times.So, that'd be two squared.So, 12 two.And then I had my calculator.I said, "Well, I really need another point."So, I did one.And at one, you're already negative at about 1.5.1.6.So, I must have crossed somewhere probably in here between one and two.And then went up.If I fold in my paper, and I did this on graph paper, what would happen if I fold in my paper?What do you think, Daniel?
What do you think happens to these two if I do it?
Then they would land on top of each other.
Yes, that's exactly right.That's exactly right.So, this is just a reflection across the X equals Y line.And this is stuff that you probably already knew, right?Or had surmised.
This chapter contrasts direct and inverse variation using formulas and graph types. It then works through example problems involving wire weight and gas laws to practice finding constants and solving for unknowns.
So, let's talk a little bit about more about direct and inverse variation.Um so, if you have direct proportion, your Y is just some constant of proportionality times X.This is a constant of proportion.Thank you.I don't This is like off first.There.Maybe that's the problem.This is called the constant of proportionality.
So, there's some constant you multiply by and another thing that we know is um uh y over x.I'm going to write that.
is always equal to k.When you divide them, you're always going to get k as long as x is not zero.Inversely, um you have x on the denominator.So, x is uh proportional to 1 over So, this is xy equals k.When you multiply them together, you get k rather than when you divide them.
So, this is going to be a hyperbola, isn't it?And isn't this just a line?Slope is I mean, the y-intercept is zero.It's just a line.But, this is a hyperbola.Let's look at an example.Make it a little bit more interesting.The weight in grams of a piece of wire is directly proportional to its length in centimeters.So, weight is directly proportional to length.If it weighs 100 g and is 25 cm long, what is that?
What times 25 is 100?Four, right?Now, find the weight of the wire if the the wire is 60.So, I need to do 4 * 60.I have to do this in two steps cuz I need that constant of proportionality before I can transfer it on.And then, 4 * 6 is 24, so this is 20 240.And it's in centimeters.The constant, we don't really usually give it any units.Let me write that one.And then you guys know your gas laws, right?So the volume of gas varies inversely with the pressure.One of the gas laws.
So the more pressure you have, the the less volume it takes up, right?That makes sense.So if the volume is 600 cubic centimeters and the pressure is 25 mm of mercury, um what would my constant be?Well, let's see.Oh, and this is on the bottom.Make sure you're in the right place.
So I would multiply.Let's see.If I had six quarters, I would have $1.50.
And two more zeros.
Where did you get the 600 from?
Did I Uh 64 Oh, 6,400.I I don't know.I have no idea.I made a 6,400.
Good question, Daniel.I have no idea.I must have been thinking about something else.So now I need um because I have this is inversely and my pressure is on the bottom, I'm going to have to multiply these two.And I get let's see, 6,400.Right?times 25.I get a big number.I get 160,000.
24 Do I have the right one?2 3 4 Yes.So now it asks me I determine the volume when the pressure is 20.So I need the volume.I have this.And now I have 200 uh millimeters of mercury.So I can divide by 100.And I get uh 1,600 divided by 2 Uh let's see.That would be what?16 divided by 2 is 8.So it looks like 80.Is that what you get?And then I need to say what this is.Uh this is centimeters cubed.
So if the pressure is a lot more I don't get as many centimeters of gas, right?The more the pressure, the more it compresses the volume of the gas.So inversely proportional.
Okay.
How are we doing on time?I thought we could just do this one in class and you can give it to me.And then people online will have to do it all by themselves.But we can check back in and see how you're doing on your answers.This should take just a couple of minutes.And I'm not going to pause.They can fast forward cuz I've got to pause.I've got to turn it back on.Okay, let me bring you one.
I got the back page if you have it, but
Oh okay, switch it with me.
So just um this is a homework, so make sure you give it to me.So do you have uh three pages?The front and back and one more page cuz I have an extra back page so Okay, so this is yours.It's like, where is the other one?Here, just switch with me so that you don't have to Try to Oh, okay.Good luck with that.Let's start.I think
Okay, so here we go.
I'll do it, you do it, and we'll see if we agree.
Check in.What did you get for lunch?And I think some of these are not worded correctly.I had to change them.So, I said the first one was linear, did you?What did you say the second one was?Quadratic.And then this is not exponential cuz they didn't say anything about a a factor, right?So, I would say proportional and I would even add the word directly proportional and why they said and is not right.A directly proportional when they just say proportional they usually mean directly.So, one is equal to a constant times the other.And then in the next one, if the product of the of the two variables is always constant, remember I showed you that?If you have y equals 1 over x multiplied by x, xy is equal to the constant.So, that's inversely proportional.And then I just said polynomial.
Do you agree with me on number five?
So, sometimes the word banks are like not that helpful.And and even these are not.I had to change that to A and I had to change that to A.Okay, so number one, I got linear and quadratic, did you?Yeah?Okay.Did you get How far did you get on the next page?Did you get all the way?
Number five.
Number five, okay.Let me catch up with you cuz I was messing up.
The speaker checks answers from a worksheet and discusses how to identify linear, quadratic, proportional, inverse, and polynomial relationships. A real-world exponential example with bacteria growth is also introduced.
All right, did you get B?Did you get C?
Did you get B?X Y is a constant.Okay.So, describe a real-world scenario where a linear function might be used to model the situation.
Okay.
I put The of money somebody makes from a job without raises or bonuses.
How about by the hour?
I'm going to put or over time.Okay, I like it.That works great.I was going to say um the number of um repetitions I can do on a particular machine weightlifting machine um it goes up by the semester.I'm not real fast.I go by the semester.Some people go up weekly, but I only do it twice a week, so.And then um it talked about how inversely proportional is different from directly.Inversely, when you multiply the variables, you get a constant.For directly, when you have the ratio of the variables, you get a constant.So, that's kind of important thing.And that was referred to on the other page on the front where you had to answer wrong.This is not exponential.This is not quadratic.
And then give an exponential function in a real world context and explain its significance.Can you give me one?Can you give me the first example you can think of?
With the growth of bacteria.
Yeah.
Um I don't know.I'll say we start with 200.
I don't know.I I have no idea what the growth rate of this bacteria is, but we're going to say that.Would you would you like to me to change any of the numbers?
Want to start with a different amount of bacteria?Okay.No, 200's good.Um bacteria actually probably in the millions, but bacteria is so scary.Bacteria can it it
it expands as it grows and you would be surprised how fast bacteria can get out and infect other things.Wow, it's pretty scary.Good example, Kayden.Thank you.All right, so um I think I've I've beat a dead horse on with this.So, um I think I only gave you one other thing.Did I give you the inversely proportional worksheet?I don't remember if I did or not.Maybe.Um no.Okay.So, you've already done your homework today for today um with this worksheet that we just did.This was the homework for today.So, those people that watched the video will probably get it right and those people that didn't watch the video probably have a crash and burn session like um So, let's go back over.
The lesson returns to standards, covering univariate versus bivariate data and discrete versus continuous data. It ends with final exam logistics, timing, and a brief closing conversation.
There is a whole Kuta worksheet, but I don't have it due for a while, so I'm not going to have it due yet.I've been working you pretty hard, so I'm going to come back off a little bit.So, I want you to go back to where you were doing the standards.
I can find mine.Let me go back to the technical stuff there.The standards.And we pretty much talked about everything, but now we can check these off, right?You got this, inversely and proportionally.Directly and inversely.So, here it's the ratio.And here it is uh the product of the variables.If you want to think of it that way.Um univariate and bivariate data.Remember, sometimes we have an X and a Y.So, like um how far I've gone based on uh time in my car.So, that's a bivariate relationship, time and distance.But, what if I just have one single variable that I'm looking at?Maybe like the number of pieces of chocolate I ate yesterday.Not good, right?I'm not going to tell you.It'll be dis- disheartening for me.So, you know, maybe the number of pieces of chocolate you eat in a day.
That would be univariate data.Just one variable, number of pieces of chocolate.So, one variable
and then this one would be two variables.
Now, um we have two ways that we can get data.It can be discrete, like I usually just eat the whole piece of chocolate.I don't nibble on a piece of it, you know, and then come back, you know, especially if it's those little ones.So, discrete would be countable.That's what discrete is.I can count it.
And continuous would be more of a measurement.
Like your height is a measurement.And we don't have to have countable numbers like 1 2 3.You can be 6.57
8 ft tall.Right?Depending on just how finely you want to measure it.Um if I want to measure a a height above the surface of the place where I'm at.Um that would be a continuous measurement.So volume, continuous.Height, continuous.Um area, continuous.Right?So all the things that you measure, we think of as continuous.All the things that you count, we think of as discrete.So the next thing that we're going to do after we finish this area with um inverse and direct proportions and a little bit more on types of functions, we're going to go to um histograms, box plots, frequency distributions,and some of the measures like mean, median, mode, standard deviation, variance.We're going to talk about those.And then guess what happens after that.Final exam.So exciting, right?Final exam.Not exciting?You guys know this.You know this up top.
Okay, I know you do.So you I'm sure you'll do well on the final exam.The final exam will be here on a Tuesday.I know.Monday.Monday.Sorry.And I even have a time.I just don't know where cuz put us all over the place.It's not necessarily in this classroom because another person might be assigned to it.So, I do not know the location.But, I can tell you if you want to put it on your calendar what the time is.I got to look at what I wrote it down.
10:15 to 12:15 on Monday the 7th of December.All right.So, 10:15 to 12:15 on Monday of finals.It is uh the 12th of Uh sorry, 7th of December.12/7
So, when are final week
Um it's Nov- it's December 7th through the 11th.
Okay.
And it you have to look up Well, you teach your instructor should tell you.Professor should tell you.I mean, you shouldn't look it up, but they might.There's a schedule in in um Canvas that you can look up.Well, I know it's not even in Canvas.I think it's just um like when you sign into the portal.
Okay.
You know all that stuff.Um there should be a final exam schedule.But, if you tell me a class, I'll tell you what it is.
Okay.
It's based on the day of the week that it meets and the time.So, since we do Monday, Wednesday, Friday at noon, our final exam is at 10:15 to 12:15.But, if you want me to look something up for you, I'm happy to do it cuz I I know where it is.It'll be fast.Stop.
This is my only in-person class that I have a finals for.
Ah.So, I'm the only meanie, right?I would feel bad about that, but you know what I think?I think I need to put pressure on you guys so that you'll be great even better teachers.I know that you're going to be great teachers, and and the best I can do is try to make it really
And I think my timing for my other two in-person classes are the first week of December.I think it's like that.
So, they give you a week?
Yeah.
Yeah.Here's is going to be 2 hours wherever room they assign us.I'll let you know as soon as they tell me.Um but it'll be a 2-hour final.And then you'll be done.You you don't have to keep going.
Yes, and then it's vacation time.
Yes.Do you get vacation or do you guys still have to practice and all that?