Exponential function basics common core algebra ii homework

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California Baptist University

Robert B.

Algebra

10 months, 2 weeks ago

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Find equations for exponential functions that pass through the pairs of points given.$ (Round all coefficients to 4 decimal places when necessary.) HINT [See Example 2.] Through $(1,1.2)$ and $(3,0.108)$

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So sometimes they're not given nice data tables and told what the value is when X equals zero. So we have to use some other methods to find it. So let's consider if we were given to 36 as a point and then For 324 and don't get too bogged down with the actual example because this is going to be applicable with all the problems, there's a set of steps that we want to follow. So notice that we're going from 2-4 And then from there we're going from 36 to 324. But remember that with a um an exponential function, you're multiplying by the same value. So this means that we have when we're up to we have some value and then we multiply it by our exponential base to get three at some point and then we multiply by exponential base again to get four and 3 24. So what that tells us is that we want 36 times whatever our exponential bases times itself again. So really just x squared And that's going to end up giving us 324. When we solve that we end up getting that X is equal to three. So really what this exponential model is going to be is why equals 3 to the X. However we want to make sure that it intersects with this point. So in x equals two instead of Equalling nine in the sequel 36. So it's going to be four times this, it's that way we ensure that it hits our point here and then it also hits our point right here. So that's going to be our final answer. And that's the process we need to go through.

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Hello, my name is Kirk weiler, and this is common core algebra two. By E math instruction. Today, we're going to be doing unit four lesson number three on exponential function basics. Most of what we're going to do today will be review from common core algebra one. But given that you had that year of geometry in between the two. We think that it's important here at E math instruction that we get you up to speed on what an exponential function looks like. How it behaves and how you can tell that behavior by simply looking at the equation. So let's jump right into two graphing exercises. That really show how exponential functions behave. But first, let's take a look at their general form. All right. Exponential functions, Y equals B to the X, where B is the base. Remember, these are functions like Y equals two to the X Y equals one third to the X, et cetera, right? So two to the X 5 to the X, half to the X .75 to the X, right? All of these are exponential functions because the base is a constant and the exponent is the variable. Note that in every single case, the base is positive. You will never have a negative base as an exponential function. You'll never have something white, Y equals negative three to the X, all right? That will never happen. But the base will always be positive. It could be greater than one. It could be between zero and one, but it will always be positive. Let it clear out that little bit of writing. And let's jump right into it. All right, exercise number one says consider the function two to the X fill in the table below with out using your calculator. And then sketch the graph on the grid provided. So I would like you to try to make an attempt on this, okay? Pause the video, take about 5 minutes to try to graph one of the most basic exponential functions. Y equals two to the X. All right, let's do it. Well, let's start with the easy ones. When X is one, two to the first is obviously two. When X is two to the second is four. And when X is three, two to the third is 8. Now notice in each case what we're really doing is we're multiplying the previous output by two. Therefore, you could go in the opposite direction by dividing each output by two. Of course we should know that two to the zero is one. And we should also know that two to the negative one is one over two to the first, or one half. Two to the minus two is going to be one over two squared, or one fourth, and likewise, whoops, find a little space down here. Two to the negative three will be one over two to the third, which will be one 8th. So let's plot these. Again, I'm going to start with the easy ones, zero, one, that's a pretty simple one, two, four, one, two, two, four, three, 5, 6, 7, 8, now for the fractional ones. Negative one, one half. And one fourth and one 8th. So we'll get something that looks kind of. Like this. Years ago, I had a student in calculus called this the hockey stick graph. Right, it goes up very quickly, and then levels out towards the X axis. So there it is. Y equals two to the X we should be able to handle those negative exponents. So it's positive exponents. That's zero exponent. We could even have some fractional exponents in there too, but who would really want to deal with two to the one half, right? The square root of two. Anyhow. Copy this down, pause the video now, and then we'll move on to one where we use our calculator. All right. Clearing it out. Now Y equals two to the X might be the most basic exponential function where the base is greater than one. Let's look at an exponential function where the base is less than one, where the base is one half. Now here it asks us to use our calculator to help us fill out the table and sketch the graph on the axes provided. So let's do it. Let's bring out the TI 84 plus. Ready? All right. There it is. We're going to just create a table of values and it's going to help us out. So let's hit Y equals. And let's put that function in there kind of the way it looks. Let's put the one half in parentheses. You could of course just put .5, but if it were a fraction you weren't as familiar with, you know, you'd want to put it in parentheses. So that's what I'm going to do. But one half in parentheses. And then I'm going to raise it to the X, make sure you do that. Don't just have one half times X, that's a linear function. So now I've got one half raised to the X now I need to go in and do a table setup, right? So let's go into our table setup. Let's make our table start at negative three. And let's make it go by ones, right? Because that's what my table goes by. All right, check that out. Make sure it all looks good. And then let's go into the table. Which we can access right here. All right, we're now into the table. So now it's just a matter of writing things down. So for negative three, what we see is we see an output of 8. For negative two, we see an output of four. For negative one, we see an output of two for zero, no grace prize we see an output of one. For one, we see an output of 0.5, which is of course just one half. For two, we see an output of 0.25, which is one fourth. And for three, we see an output of 0.125. That's less common, but that's one 8th. So let's graph it. Negative three, one, two, three, four, 5, 6, 7, 8. Negative two, one, two, three, four. Negative one, one, two, zero, one, one, .5, two, one quarter, and three. One 8th. There's probably no way I'm going to draw a good smooth curve through this, but I'm going to take my best shot. You guys all know how bad I am at this. And there we go. Now, of course, that exponential function decreases because it's base is less than one. You know, and it makes sense. Every time we multiply something by one half effectively, we're dividing by two, so we're making the quantity smaller, in that case. All right? I'm going to put away the TI 84 plus for right now. We might need a little bit later, but let's get rid of it now. All right. Pause the video, copy down anything you need to, and then we're going to move on. Okay. It's clear it out. Okay. So exercise three says based on the graphs and behavior you saw an exercise one and exercise two, state the domain and range of an exponential function of the form Y equals B to the X see, the thing is, when you have Y equals B to the X, right? It either looks like this, if B is greater than one, or it looks like this. If B is less than one. But its domain write the input set, well, that's all real numbers. There's absolutely no value of X, you can not put in there. I mean, we now can handle zero. We can handle positive exponents negative exponents fractional exponents. But the range. That's a little bit different. Notice that in every case the outputs we always got were positive numbers. Positive outputs we never got an output of zero, and we never got a negative. And that will always be the case, the range is Y greater than zero. If you really like interval notation, it would look something like this, zero to infinity. All right, but not including the zero and not including the infinity. All right. This is pretty important stuff. So pause the video now. And write down anything you need to. All right, let's clear out the text. Our exponential functions one to one. How can you tell? What does this tell you about their inverses? All right, pause the video now and think about this. It hasn't been that long since we've done one to one functions. So think about whether exponential functions are one to one. All right. Well, remember what a one to one function really is. One to one function means that different X's, whoops, that different X's give different Y so no wise get repeated, right? Well, if you think about their graphs, whether it's increasing exponential graph. Or a decreasing exponential graph with a random mark there. Because they're always increasing or always decreasing, they are definitely. They're definitely one to one. You could also use what's known as the horizontal line test. Remember that passes? The horizontal line test. And remember how the horizontal line test works, right? It says that if you take any horizontal line, let me draw it in red. And it hits the graph at most once, right? It never hits the graph twice, then it's one to one. Now possibly the most important thing, what does this tell you about their inverses, it tells you that their inverse. Is also a function. Or perhaps grammatically speaking, I should say the inverses are also functions. All functions have inverses. The question is whether those inverses are also functions. And exponential functions that will be the case. Now, we won't see their inverses for a little while. But they're going to be very important part of this course. So pause the video now and write down anything you need to, and then we'll move on. Okay. Clear out the text. Let's take a look at another one. Exercise 5 says now consider the function Y equals 7 times three to the X let a race is determine the Y intercept of this function algebraically, justify your answer. Okay, fair enough. Well, why don't you go ahead and try to find the Y intercept algebraically. See if you can remember how to do that. All right, let's do it. Well, to determine the Y intercept, any function, right? I don't care what it looks like. Plain fact is for the Y intercept, the X value is zero. So let's see what we get. Y equals 7 times three to the zero. Now remember your order of operations, that multiplication by 7 comes after we raise the three to the zero power, but three to the zero. Is one. So we just get 7. That is our Y intercept. Isn't that cool? That's our Y intercept. So letter B, does the exponential function increase or decrease explain your choice? Well, think about this for a minute. Ah. This exponential function increases. Because its base three is greater than one. All right, so it's all about whether the base is larger than one or less than one. Please don't get wrapped up as some students do thinking about it's whether or not the base is fractional or a whole power. Not a whole power, but a whole number. That's a terrible way to look at it because if I have a base of 5 halves, even though that's a fraction, the 5 halves being greater than one will mean the exponential function will increase. Now, let her see it says create a rough sketch of this function labeling its Y intercept. Here's the thing. All exponential functions that increase essentially look like this. The only difference really is how fast they increase and where they cross the Y axis. But that's it. That is a perfectly good graph of this exponential function. All right. Well, I'm going to clear this out in just a bit. So pause the video now, and then we'll move on. Okay. There we go. All right, exercise number 6. This one's a bit different. Consider the function Y equals one third to the X plus four. How does this function's graph compare to that of Y equals one third to the X? In other words, what does adding four due to a function's graph? Think about that for a minute. All right, let's talk about it. It will shift or you could use the word move. It will shift the graph of Y equals one third to the X upward. By four units. Let her be. Looks like we've seen this one before. Determine this graph's Y intercept algebraically justify your answer. All right, go for it. All right. Take a look. Again, Y intercept. Means X equals zero. So let's just put that in. We'll get one third to the zero plus four, one third to the zero is one plus four gives me a Y intercept of 5. But our C says create a rough sketch of this function labeling its Y intercept one. This is a decreasing exponential function. But normally, one third to the X would look something like this. I'm going to draw it in dashed. This would be what one third to the X would look like. All right, so this is Y equals one third to the X and it would have a Y intercept at one. But what we've done is we've shifted everything up for units. One, two, three, four. Draw another dashed line. I'm going to also put that 5 on there. Because this is what our new function looks like. All right. Instead of getting whoops sorry about that, instead of getting smaller and smaller and approaching the X axis, it gets smaller and smaller, and approaches the horizontal line, Y equals four. All right? So we've shifted our exponential function upward so that it is no longer leveling out at the X axis, but leveling out at a different horizontal line. All right, pause the video now if you need to. And then we're going to clear this out and finish up. Okay, very good. Here we go. All right. Well, in today's lesson, we probably reviewed a lot of things that you saw in common core algebra one, probably. We didn't see anything new at all. What we wanted to do was review basic characteristics of exponential functions, the most important one, being the fact that the base and whether it's greater than one or between zero and one dictates whether the exponential model is increasing or whether it's decreasing. This is very similar to what the slope of a linear model tells us. Slope of a linear model is positive. It increases if it's negative and decreases. Here, when the base is greater than one, it increases when the base is between zero and one, it decreases. And then there was a lot of other stuff thrown in there as well. All right. For now though, I want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler. And until next time, keep thinking. I keep solving problems.

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