# 4.8 Fitting exponential models to data  (Page 5/12)

 Page 5 / 12
$y=\frac{c}{1+a{e}^{-bx}}$

Recall that:

• $\frac{c}{1+a}\text{\hspace{0.17em}}$ is the initial value of the model.
• when $\text{\hspace{0.17em}}b>0,$ the model increases rapidly at first until it reaches its point of maximum growth rate, $\text{\hspace{0.17em}}\left(\frac{\mathrm{ln}\left(a\right)}{b},\frac{c}{2}\right).\text{\hspace{0.17em}}$ At that point, growth steadily slows and the function becomes asymptotic to the upper bound $\text{\hspace{0.17em}}y=c.$
• $c\text{\hspace{0.17em}}$ is the limiting value, sometimes called the carrying capacity , of the model.

## Logistic regression

Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command “Logistic” on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form

$y=\frac{c}{1+a{e}^{-bx}}$

Note that

• The initial value of the model is $\text{\hspace{0.17em}}\frac{c}{1+a}.$
• Output values for the model grow closer and closer to $\text{\hspace{0.17em}}y=c\text{\hspace{0.17em}}$ as time increases.

Given a set of data, perform logistic regression using a graphing utility.

1. Use the STAT then EDIT menu to enter given data.
1. Clear any existing data from the lists.
2. List the input values in the L1 column.
3. List the output values in the L2 column.
2. Graph and observe a scatter plot of the data using the STATPLOT feature.
1. Use ZOOM [9] to adjust axes to fit the data.
2. Verify the data follow a logistic pattern.
3. Find the equation that models the data.
1. Select “Logistic” from the STAT then CALC menu.
2. Use the values returned for $\text{\hspace{0.17em}}a,$ $\text{\hspace{0.17em}}b,$ and $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ to record the model, $\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}.$
4. Graph the model in the same window as the scatterplot to verify it is a good fit for the data.

## Using logistic regression to fit a model to data

Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service. [link] shows the percentage of Americans with cellular service between the years 1995 and 2012 Source: The World Bank, 2013 .

Year Americans with Cellular Service (%) Year Americans with Cellular Service (%)
1995 12.69 2004 62.852
1996 16.35 2005 68.63
1997 20.29 2006 76.64
1998 25.08 2007 82.47
1999 30.81 2008 85.68
2000 38.75 2009 89.14
2001 45.00 2010 91.86
2002 49.16 2011 95.28
2003 55.15 2012 98.17
1. Let $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ represent time in years starting with $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ for the year 1995. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.
2. Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.
3. Discuss the value returned for the upper limit, $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ What does this tell you about the model? What would the limiting value be if the model were exact?
1. Using the STAT then EDIT menu on a graphing utility, list the years using values 0–15 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in [link] :

Use the “Logistic” command from the STAT then CALC menu to obtain the logistic model,

$y=\frac{105.7379526}{1+6.88328979{e}^{-0.2595440013x}}$

Next, graph the model in the same window as shown in [link] the scatterplot to verify it is a good fit:

2. To approximate the percentage of Americans with cellular service in the year 2013, substitute $\text{\hspace{0.17em}}x=18\text{\hspace{0.17em}}$ for the in the model and solve for $\text{\hspace{0.17em}}y:$

According to the model, about 98.8% of Americans had cellular service in 2013.

3. The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be $\text{\hspace{0.17em}}c=100\text{\hspace{0.17em}}$ and the model’s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!

I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert