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Find the exponential function that models the data in the table below:-2 383122448What is the exponential regression of the data?...

Question

Find the exponential function that models the data in the table below:-2 383122448What is the exponential regression of the data?

Find the exponential function that models the data in the table below: -2 3 8 3 12 24 48 What is the exponential regression of the data?



Answers

Find an exponential function determined by the data in Table 8 . $$\begin{array}{|l|l|l|l|} \hline 20 & 40 & 60 & 80 \\ \hline 100 & 72 & 51.84 & 37.3258 \\ \hline \end{array}$$

So we're giving this data set here with our X. Values and our F. Of X values. And the first thing that I like to do to figure out if this is a linear um data set. Or if we can make a linear model fit this data set. As I like to look at the difference between our f. Of X values when we're going by the same increment in terms of our X values. So if we're going up by one which we are in this status set and I like to look at the differences between each of our X values when we're going up by our fx values when we're going up by one X value. So in this case three minus three halves is three halves 3 -6 or 6 -3 is three. 12 -6 is six and 12. Or sorry 24 -12 is 12. And so for this to be a linear model we would have to have the difference between each of our f of X values be equal to each other. We've had been going up by the same amount each time we're going up um by one x value. So this isn't a linear model. But now what we want to check is if this is an exponential model, so the way that I like to go about doing this is I like to look at what are F X values are when X is equal to zero and when X is equal to what? So if this is an fx or if this data set can be fit to an exponential model, then our F. Of X when X is equal to zero is going to be equal to. So f of zero, it's going to be equal to something um raised or sorry, it's going to be equal to a which is just some constant raised to the zeroth power and then multiplied by some constant C. And so at f of zero, this is just going to be equal to see. So in this case C would be equal to three. So we know that if this is an exponential model, C is going to be equal to three. And then the next one that I like to look at is X equaling one. So if we took F of one where this is an exponential model that fits our data set, then we would have a which is the constant that raised to the X power to the exponent. Um And this would be raised to the first power since this is F of one and then multiplied by C. Which we already figured out is three. And we know this is equal to six. So what we can do is we can divide by three. We can say A is equal to two. And the reason again we know this is because this is a race the first power so A would have to be equal to two. For this to be an exponential um refer an exponential model to fit this data. So the exponential model That would fit this data would then be three times to raise to the X power. And we can see that this is actually going to be the model that fits our data. We plugged in negative one, you would have three times two to the negative one or three times one half which is three halves which is what we got. And we already know that at three and six we have the right um exponential model. And then at two we can see that we're gonna have to squared times three which is 12 which is what we have for Fx. And at three we're gonna have to to the third, which is eight times three, which is 24 which is again what we have when X is equal to three. So this model um f of X is equal to three times two to the X is the model that will fit this data, and it's exponential model.

So we have this data set and we want to figure out if ffx can be fit as a linear model or an exponential model or neither. So the first thing I'm gonna do is look at the differences between our ffx values given that we're going at a constant increments up in the X. In terms of our X values. And so we have one minus two thirds which is one third 3/2 -1 which is 1/2 9/4 -3/2. And then 27/8 minus 9/4. Well 9/4 is 18/8. So that would be 9/8 mm. And so we can see that we don't have a constant increase in terms of our F. Of X values. And that means that this isn't a linear model. If we had a constant um increase or the difference between our F. Of X values was the same um Between each of these F. Of X values, then we would have a linear model but we don't so we don't have a linear model. So now what I like to do is since we're given X at X is equal to one. If this was a exponential model it would probably be three over to to the X. Since we have X is equal to one is 3/2. And we actually can um confirmed that sense at zero X is equal to zero. We have F. Of X is equal to one. So let's say F of X is equal to some constant A times three over to the X. Well if F of zero We know is equal to one that we know that eight times 3/2, 20 powers equal to one which is equal to just a. So they would have to be equal to one. So in order for this to be an exponential function, it's going to have to be 3/2 to the X. Since at X is equal to one, we have 3/2 0. We have X. F. X equal to one. So now what we can do is we can just make sure that's true. So 3/2 squared would be three squared which is nine divided by two squared is four. So this is correct. Three to the third is 27 2 to the third is eight. So this is also right and then three over to to the negative. First power would be the reciprocal of 3/2 or two thirds, which is also right here. So a exponential model for this data which would fit the data is Y. Is equal to three over to to the X.

In this question, the exponents L model is given by y is equal to a to the power BX. Now I am writing the value of X and y here. So the value of X and Y is 03 and here it is one and one by four. So at point X is equal to zero. Why is equal to one at point 01 I can like they have of expression is one is equal to e b two b multiplication zero. So I get the value of H one from the other big question Now at point access equal to three. And why is equal to one by four with a as one, I can idea. Publication edge one by four is equal to 18 to the power bi multiplication three. On solving I get three b is equal to lawn one by food so physical to minus 1.3863 by three, which is equal to minus 0.46 to 1. So the exponential model equation is given by Y is equal to eight of the power minus 0.46 21 ads, which is our answer for this problem

Question 25 is asking you to find the experimental model that fits the points shown in the graph or the table for Question 25. You have a graph, and it states two points que 01 and be 3, 10. Um, this is an exponential growth. You can see from the graph, which has the formula y equals a E to the B X. And we can use these two points to find this. Um, so starting it X equals zero and y equals one. Pulling that into the equation, you get one equals a e to the B time. Zero e to the zero is equal toe one. So from there, you know, a is equal toe one using our second point X equals three and y equals 10. You can plug those points and you get 10 equals one, which is a e to the B times three. From there, take Ellen on both sides, and then you can get rid of our e value here. So you get Ellen of 10 equals three. Be solving for B. You get how one of 10 divided by three. So plugging that into our equation to get the full, um, model you get. Why equals E to the Ellen of 10, divided by three multiplied by ax and that is your expression for a question 25.


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