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Exponentials PART 1

Added Jan 10, 2023 •

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Exponentials PART 1

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what's going on y'all is your math teacher Miss Harris and I'm back with another video so disclaimer before I even get started I had previously already posted this exact video on Karam barely my microphone was disabled yesterday when I did the video so I am so sorry I'm going to have to re-record this once again but I just want to let you guys know we are officially done with a cute saying but yes we are done with the sense of equations and now we are moving into exponential functions so you already know what to do get your notes get a pencil get something to practice with so when we get these practice problems you guys already are prepared and let's get into it so first things you already know it's going to be where while so I need you to pause your screen right here shortly and I want you guys to go ahead and write down the five worst I have here exponential function exponent appreciate compound interest and principal amount the word and its definition in the will go ahead and move on thank you all right so hopefully you guys got those words written down the words and the definitions and soon very soon I will upload the quarter forward while it's for you guys if I have not done so already but I know some people like to have it on paper and just highlight what we got you know as we move along through these accused but if you're one of those people that like to write it down feel free to keep doing it okay let's go ahead and move into today listen all right so what I want you do I just want you to watch what I am doing and I want you to think about what you are noticing what you are observing what are you wondering okay and am first thing I'm going to do is I'm going to grab these following points on find My Graph right here okay I'm going to grab one time and one I'm going to grab two, two I'm going to grab three, four and I'm going to grab four, eh all right so here we go and just watch what I do and I want you to think about what you are noticing by let's start with one, one here we go I'm at 1 on the x-axis one on the y-axis there's one coming one if we go to, too it was all three, four and here we go for, 8 okay so just want you guys to notice what you got right here and I'm going to stop right there so what do you notice what do you want about this graph now I'm going to fill in my table okay now before I can fill in my table I got to figure out what pattern is already going on like what am I doing what operation am I doing am I adding and subtracting multiplying and dividing and by what number it has to be the same number over and over and over again okay what what I just noticed off the bad looking at the Y column I know that I'm going from 1 to 2 so I'm actually a thing okay I'm heading one check is 1 + 1 gives me then 2 to 4 okay now I got to add to them for the eighth okay now I got to add for work I'm adding every single time but I'm not adding by the same number so you cannot be my pattern okay so let me go ahead and write this and then think of what else could be going on now what I noticed is that I have one 248 so on and so forth so continuously as I go through this chart my numbers are increasing their getting larger and larger song I know the only operations that get larger I got to be adding and multiplying so that's that's what we're going to go with next let's try most fine what how do you get from 1 to 2 with more time you think you're going bye-bye to try to get from 24 by multiplying easy I'm about to and 48 how you going to get there by multiplying Easy by to buy to every single okay so I'm going to do the same thing in order to fill in the rest of time Sam Tsui that's going to give me 16 that's my missing number here okay I don't know what I'm going to have to do here but I'm going to assume that I'm going to have to work backwards because if I don't know what number it is I don't know what number times to was going to get me to 1 I don't know what what number times she was going to give me to that next one so I'm going to have to work backwards so instead of x to I'm going to do the opposite which is divided by 2 so I'm going to do is I'm going to do 1 / 2 which I know is going to be one half then I'm going to do one / 2 cuz wasn't working backwards and 1/2 divided by 2 or 2 / 1 is the same thing is that Keep Change Flip that you guys heard about okay where you keep that first fraction you change from division to multiply and then you flip that second fraction to a typical so now you just going to go straight across and got one x one with you if you want into * 2 which gives you or so that's what's missing and the tables I want you guys to think what do you know at this point what are you wondering at this point like what can You observe from the graph that I just created and the way I just filled in this paper okay what I also need you to think about it I want you to go back like a while back like 23 all the way back member domain and range remember family functions where we made that chart this is what we're going to be focusing on today where we kind of bringing that back into our exponential function so if you haven't already caught on with our consistent pattern of multiplying over and over and over again are numbers getting a potentially larger and the fact that my graph is curving here as soon as I start plotting points I clearly have an exponential function and you're going to see that over and over and over again today in this video all right so here we go. Let's move on let's talk about the main formula that you are going to need you are going to need this bad boy right here y equals 80 * B to the X power wife was a times B to the X power I can not say that enough in the video this is the number one thing you're going to need in order to be successful with exponential functions okay let me tell you a little bit something about why our guys just don't they the same now you're a I know you haven't really now before I beat your  is going to represent your Y intercept your initial value the number where you start at okay that's what they represent in the next Niger function B is going to represent your base number is going to represent your grows back there is going to be how your number is either increasing or decreasing okay and then ex is just going to represent your exponent for you simple XX Bonnie is going to be where you start be is going to be your base or your growth factor okay and it represent your ex now there's a little thing that we have to keep the Civic about A and B well a kid not equal zero and not equal zero okay and then be in order for me to be an increasing a grow a growth factor you know positively getting larger and larger larger it has to be greater than 0 and it cannot be equal to one the reason why I cannot because one of my vehicles one let's just say that I had one to the third power what does the same thing 1 * 1 * 1 which is the end of the day is still going to give me one if I had one to the 17th power that's riding 1 * 117 times and I'm not going to do all that because I guess you guys can go ahead and figure out that at the end of the day when you my father once together you're still going to get one so your base cannot be one because your function will not get exponentially larger it's going to say exactly the same because it's going to remain that same number which is 10K and if it is Ducane if it's decreasing your number we will have to be bigger than 0 but less than one so it's going to be a decimal is going to be somewhere from 0.01 all the way to 0 .99 okay so it is increasing is going to be bigger than 0 but I would have won if it's decreasing it's going to be bigger than 0 but less than 1 so I need you to do both of these boxes that you said okay so that way we can go ahead and move on and that way you got to have these notes back to when we start practicing all right go ahead and pause the screen and write this stuff down here we go let's keep it moving so once again just remember a is where you start B is your gross back your index okay alright so let's work on example number one that was going to happen is I am going to fill out the table first to show you guys how your exponents work which is something that you guys should be familiar with since we worked on in 8th grade and then I'm going to show you how you apart the dots on the graph to show that it's creating an exponential function okay X power now I know you guys like oh my goodness mysterious already starting off with fractions that's going to be insane but it's not it's really really nice and it's going to be okay let's take what we have already done for example so we got y equals 2 to the X power right what's going to happen if you got to to the -2 lb going to play that into to the negative 2nd power now what has happened is with your exponents remember you cannot have a negative Tone If you have a negative exponent you're going to have to lip your base to it's reciprocal and then your exponent will be positive okay so remember that that too is the same thing is 2/1 so when I flip it so I can make my Xbox and positive it's not going to be two or more is never going to be 1/2 second power well I know that 1/2 is that your power is the same thing as 1/2 x 1/2 day and I know but it 1/2 x 1/2 is going to give me 14 which is how we got here okay when do the same thing with negative one plugged in Forex so you got y equals 2 to the negative 1 power once again we're going to flip to or two over once it's reciprocal which is one half so now you got 1/2 to the 1st power 101st hour is just as a number that you're working with the Y equals 1 half which is how we got here okay and that's really it does all the facts that we have now to EZ park and go ahead in the race so I got some more room is when you get an exponent of 0 dad is the easy part because any numb that has an exponent of zero is going to equal one and say that again any base any number that has an exponent of one so you have like what's 2 to the zero power equal 1 any number don't care if it's a million a million to the zero power is one 500,000 to the zero power is one Infinity to the power so anytime you have an 80 please automatically know I don't have to do any work already that equals 1 okay what about when I plug in one so y equals 2 to the 1st power why don't have the most funny thing to suit the first hour is just too it's just a number that you start off with just your base number Saudi Road exponents are probably the easiest to work week cuz you really don't have to do any work you literally just look at what you have and that was your answer OK Google to to the power of 2 equals 2 square which is the same thing as two times to everybody knows that you * 2 for all right and then when you plug in 3 y equals 2 qt which is the same thing as soon * 2 * 2 which gives me 80k now I can start plotting these points oh here we go so we're going to play 0 1 2 and 3 on there that way you guys can get a good feel when you start playing in those negatives and he's starting to put those prices and gets kind of tricky I'm just going to show you what the whole numbers okay so when x equals 0 here I am x equal 0 why is going to equal 1 oh my goodness we hit the grass when x equals 1 and we got one why isn't equal 20 my goodness we hit the graph again when x equals 2y is going to equal 1 2 3 4 oh my goodness we hit the graph again and then x equals 3 here we go 3 why is going to equal 8 and I know these lines are kind of thick so it looks like it's getting more pieces of the graph but that's why you want to fight your points just to make sure that you're drawing at graph directly okay The Narrows look at our domain and range remember domain and range from eq3 always told you when you're looking for the dumb and you're going to go from the left side of the graph to the right side if you're going to take your pencil or whatever you have you're going to fly from the left you hit the arrow you that is negative Infinity you're going to keep trying and keep trying to keep silent until you hit the last piece of the graph on the right side which is I'm not getting into another arrow which is going to represent positive Infinity so negative Infinity deposit Infinity is going to be you're dumb and another way that we can represent that is all real numbers okay start the main is all real numbers you're going from bottom to top okay so when you go from bottom to top what's going to happen is is that you guys are going to have to think about what piece of the graph am I going to hit you hit a piece of the line which is at oh so that's going to be the start of your range and then you're going to keep going up and up and up and up and up until you hit the last piece of graph which is going to be an arrow which has been deposit Infinity so you're going to be from 0 to positive Infinity okay I would look at example number two example number two how does the race come on you all right example number two so he needs and once again finish the table and then we're going to completely graph these points that way you guys can see what the exponential function is going to look like now please do not get upset because you see this fraction I promise you most of these fractions it's really not that bad I promise I promise I promise let's start with these negative exponents that where you guys can refresh and what you're supposed to do when you have a negative exponent so let's start with - 3 y equals 1/2 to the negative 3rd power remember that that 3 cannot stay negative so we're going to have to do is take that base which is 1/2 and you are going to have to flip it this reciprocal so now is y equals 2/1 to the third power cycling is too cute we know what's 2 * 2 * 2 which gives us eight that's how we got the 8K same thing with - 2 y equals 1/2 to the negative 2nd power which is the same thing as y a y equals 2 squared can you give me two times to which gives me 4 that's how I got the for now we're going to keep doing the same thing with - 1 y equals 1/2 to the negative first Power which is the same thing as member flip your base 1 to the 1st which is just too okay now remember anything to the zero power is out of matically one alright hidden anything to the first Power it's just going to be your base now that you start with the Starbase number is 1/2 1/2 to the first hour is just going to be one half it was an exponent of one you're just going to put Air Base number okay now I'm going to plug in y equals 1/2 to the second power which is the same thing that's why you was 1/2 x 1/2 x straight across 1 * 1 gives me 12 * 2 gives me a word so now I can fly I'm go ahead and racist real quick so I can watch this on the graph hang on all right okay that was going to happen is I'm going to start at - 33 equals -3 + -3 OK Google. -2 of a doing going to go all the way up to 4 -1 years - 1 going to go up to 2 if x is zero here's x 0 Y is an equal 1 if x is one it's going to be 1/2 I'm going to go buy halfway white and then x equals 2 I'm going to be about 1/4 on my right here row closest zero okay so now what I'm going to do is I'm going to put my line through my grab and you can clearly see that you got this initial graph but if you notice this one when it was getting bigger and bigger bigger we're going from bottom to top with this one since it's getting smaller and smaller and smaller is decreasing we're going from top to bottom okay so that's what I want you to realize when you have a growth exponential form going to go. When you have a Decay exponential function is going to go from top to bottom now let's look for the pain and the rain so I'm just going to put the right there they are all right so I'm going to go from left to right once again starts with a w first thing I hated the arrow the net the last thing I hear it is a narrow let me see my domain is all real numbers I am arranging to go from bottom to top it we going to hit the first part of my grandpa just don't zero so it still zero Broadway up in his Arrows with positive Infinity so my domain in my range is still the same but I just want you to realize the difference between your graphs okay so when it gets bigger when it gets smaller top to bottom right here we go all right so we're just comparing he's like pretty much what I just did on the first ring I just want you guys to realize that once again your domain is all real numbers your range starts at zero give me one second y'all alright sorry y'all I got a I had a whole teacher come in my room so as I was saying the domain is pretty much the same from the last line of the range is still the same from the last by the y-intercept now let's talk about this y-intercept so if you plug in zeroes that we got y equals a times B to the X and once again if your ex is zero you already know is there we know that be 20 is going to be once a y equals 8 * 1 which is still I give you a Sammy's the y-intercept is going to equal a that is why this ordered pair is 0, a okay then what's it going to be is bigger than 1 year graph is going to increase when B is bigger than 0 but less than once with a is going to be great okay now the grass have an asymptote at y equals 0 at the x-axis now I know you're lying because I have no idea what to ask so there's that's okay I'm going to highlight the accent or you asking so is nothing but a lie where the graph starts the car which is right at 0 that's why I said y equals 0 at the x-axis cuz here we are on the x-axis K and this is where your grass starts to Curtis where your grass starts the curb so it starts the curve at 120 on the x-axis okay all right let's go ahead and get off of this and go to the next part of our lesson today which is once again trying to practice with X and it's a function along with that formula that I gave you guys in a couple spots ago so remember that y equals a times B to the X power where a represents the number that you started as just the first number you see okay it's your y-intercept is your initial value it is where you start at the beginning okay your B is your base number is your growth factor is going to tell you whether your ex she is going to increase or if it's going to decrease in Annex of course represent your exponent okay so what we're going to do is we are going to identify our initial value we're going to identify whether we have a growth or Decay based on our Base number and then what is the actual growth factor what is the number that's in parentheses that represent That Base number okay we're going to do one and two together then we're going to pause our screen and you guys are going to do 3 4 5 and 6 and then we're going to check it alright I'm here we go let's start with 22 to the X power now once again your initial guy is the first number that you see is where you start hear you start at 33 so that is your initial value okay is it a growth or is it a Decay what look at the number of parentheses is it bigger than 11.22 is bigger than once and we know it is a grow okay in the actual growth factor what is that number that's our base that number that's just use me in the front seat Well it's 1.22 so 1.22 is going to represent our growth factor okay let's try this again y equals 12 * 3 to the 8th power start we're starting with the number Leo okay well it's b or the number to Saint Francis bigger than 1/3 is bigger than once that is going to be considered a growth what is the actual growth factor what's the number that's in the parentheses it is 3 K it's really really simple y'all some real quick I want you to pause your screen by 3 4 5 and 6 and then we will come back and we will check them and we got all right my love let's get into it so let's look at number 3 y equals 32 * 12 to the 8th power once again you're a ninja body is where you start we are starting with 32 so that would be in this is it a girl thing I will think about what number is in parentheses if it's bigger than one Instagram if it's bigger than 0 but less than one it is a Decay Point 12 is bigger than 0 but it's less than one but this one is a d a K and N what is our actual growth factor I have to go back to hear is 0.12 so awesome awesome awesome if you got that one right, let's move on to the next one y equals 1990 * 2.99 to the X power of course we're going to start at 1990 we have .99 which is technically bigger than 0 but technically less than 1 so that is a Decay even though it's right there on the line stay in our growth factor is 0.99 awesome awesome you got that one right what you want to the next one all right so why is 6 * 4 1/8 to the X power the number that we start with is 6 okay the number is in parentheses is bigger than 0 but less than once a day is once again a  head all right and then our growth factor is the actual number to sit in front seat which is 0.88 awesome job if you got that right and it last but not least Y equals 1250 * 1.2 to the X power our initial value the number that we started is of course 1250 growth or Decay inside the parentheses we got one point to which is bigger than once that is a growth a rights and then our Groesbeck going to be the number inside the parentheses once again which is going to be 1.2 so kudos to you if not it's okay you got plenty of chances to go ahead and get that together when we practice on Delta math this week okay I write the last piece of today's lesson I'm going to show you how to transform or I'm sorry nothing will transform convert whatever you want to say from a geometric sequence to an exponential function all right who's the next part of your notes I'm going to give you two examples and I'm going to have you just check yourself with two examples and of course you can have plenty of practice Sunday with the math so let's look at the first box on the left geometric converting to exponential so the number one you want to know about this money that you want to write your exponential function formula that's going to help you plug everything in that you need okay first thing I'm going to do the Y equals X a x b x and once again represents your initial your Y intercept where you start B represents your base number the number that you're you're growing by your growth factors the weather is increasing or decreasing okay for represent your exponent so I'm going to go ahead and tell you or the Y and X police say the same we're not used to playing on everything in the main two numbers are going to plug in a and be all right so you're going to play in your pattern that you see or your common ratio for be now if you remember would you measure your time and you have a Serbian equals that mean really decent me an equal Gees up 1 * R to the N -1 where are is your common ratio so you're just going to plug in where are is for B since the two is in this our position is also going to be in the bee position so now I have y equals a times two to the ex now we're going to do the next that we're going to plug it in for a you going to plug in that first term as a fraction with your comment for a so what that means is I got to plug in 9 but none has to be a fraction with two or a so now instead of having y equals 8 x 22 x and I have y equals 9 / 2 * 2 to the X power want to know why that is I have attached another video from YouTube to show you why you're having to divide 9 and 2 or just dividing use me your first term by your ratio by that works or why that is the way that it is I do now confuse anybody so if you're one of those people that are wondering that I have an extra video to show you that step and while that is okay if you're just like Miss Harrison tell me what it is and move on this is exactly what you have to do number one country formula number to plug plug in your common ratio number 3 plug in your first term / your common ratio for a and then leave X as your Xbox so this is going to be your final answer y equals 9 / 2 * 2 to the 8th power now let's look at converting exponential into geometric the once again your first step is going to write the formula that you need okay let's see here right geometric sequence so that means you're going to have G sub and equals g sub one do the n -1 power amazing things are going to plug in is G sup one and okay so once again you're going to plug in your common ratio for our let's go to be this number to Sam parentheses you're going to plug this in for are so now you got to be so and it was just that 1 * 4 to the N -1 remember Jesus and in my last Wednesday the time so the only thing I have left is to plug in my first term my first number in this sequence okay so my first time is going to be my numerator in this fraction so my numerator is this Evan right here that represents first term in the sequence so I am going to plug that in for Jeep the one and I have Jesus up in equals 7 * 4 to the N -1 power okay and that is going to be how you convert from exponential to gym alright so I do not want to confuse you I'm just going to review one more time when you're going from geometric to exponential right down the formula you need that's first plug in your, ratio for be that's next divide your first time by your kind of ratio to put that as a fraction for a delete access your Xbox that's the lift box from expedition to geometric you're going to put your number that's in parentheses your be but that is our that's your common ratio is your pattern decent one is going to be the numerator of your fraction okay to stay the same so go ahead practice at real quick and then we'll be done okay which geometric sequence below can be written as an exponent function that you got exponential function right here but we're going to convert that into a geometric sequence so I'm going to put a sub in A & G or they're pretty much interchangeable Remains the same thing as a one-time are in my power. I just need to plug in a sub one and are a step in and N -1 are going to stay the same now I remember your are is going to be the number that's in parentheses that means I'm going to plug you in 3 right here and then my AC at 1 my first time is going to be the numerator that's in my fraction just a numerator not the whole fraction to a sad one is going to be 5 so I can have a sub in equals 5 * 3 to the -1 power what is answer Choice c k all right which exponential function below represents ASAP and equals 16 x 22 in -1 power as an exponential function the geometric now we're about to change it to y equals x a x b ^ y equals f of x I need you to realize why in f of x are the same exact thing they're interchangeable okay once again we're going to plug in a we're going to plug in byx are going to stay exactly as things are going to have y equals some number X phone number to the X power okay now you're be is going to be your pattern is going to be that common ratio this to did you say that's going to go where besok this 16 is going to be the numerator in your fraction your kind of ratio is going to be the denominator in your fraction so you're going to have y equals r f of x equals 16 / 2 * 2 to the X power now I know you like me there's there's no 16 / 2 well we all know what's 16 / 2 is 16 / 2 is going to give us a 8 and then we have to do the X power once again answered or see now if this has confused you please feel free to join my live on this Friday if you have any questions if you guys need any extra help otherwise just email me you were using my email right there hair are rcschool.net text me b a remind you got Delta math that you can do the practices and once again join my live on Friday if you guys need help that is what I am here for ask any questions that you have okay I was in that thanks y'all for hanging in there with me sorry I'm having a record this video but I love you and I hope this is garnet is still going well you are almost at the Finish Line y'all just keep pushing keep pushing keep pushing I'm so proud of y'all. Have a wonderful day and until next time I'll see y'all later

Uploaded by Kristen Harris.

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Exponentials PART 1

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