# 6.8 Fitting exponential models to data  (Page 6/12)

 Page 6 / 12

[link] shows the population, in thousands, of harbor seals in the Wadden Sea over the years 1997 to 2012.

Year Seal Population (Thousands) Year Seal Population (Thousands)
1997 3.493 2005 19.590
1998 5.282 2006 21.955
1999 6.357 2007 22.862
2000 9.201 2008 23.869
2001 11.224 2009 24.243
2002 12.964 2010 24.344
2003 16.226 2011 24.919
2004 18.137 2012 25.108
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 1997. Let $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ represent the number of seals in thousands. Use logistic regression to fit a model to these data.
2. Use the model to predict the seal population for the year 2020.
3. To the nearest whole number, what is the limiting value of this model?
1. The logistic regression model that fits these data is $\text{\hspace{0.17em}}y=\frac{25.65665979}{1+6.113686306{e}^{-0.3852149008x}}.$
2. If the population continues to grow at this rate, there will be about $\text{\hspace{0.17em}}\text{25,634}\text{\hspace{0.17em}}$ seals in 2020.
3. To the nearest whole number, the carrying capacity is 25,657.

Access this online resource for additional instruction and practice with exponential function models.

Visit this website for additional practice questions from Learningpod.

## Key concepts

• Exponential regression is used to model situations where growth begins slowly and then accelerates rapidly without bound, or where decay begins rapidly and then slows down to get closer and closer to zero.
• We use the command “ExpReg” on a graphing utility to fit function of the form $\text{\hspace{0.17em}}y=a{b}^{x}\text{\hspace{0.17em}}$ to a set of data points. See [link] .
• Logarithmic regression is used to model situations where growth or decay accelerates rapidly at first and then slows over time.
• We use the command “LnReg” on a graphing utility to fit a function of the form $\text{\hspace{0.17em}}y=a+b\mathrm{ln}\left(x\right)\text{\hspace{0.17em}}$ to a set of data points. See [link] .
• Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit.
• We use the command “Logistic” on a graphing utility to fit a function of the form $\text{\hspace{0.17em}}y=\frac{c}{1+a{e}^{-bx}}\text{\hspace{0.17em}}$ to a set of data points. See [link] .

## Verbal

What situations are best modeled by a logistic equation? Give an example, and state a case for why the example is a good fit.

Logistic models are best used for situations that have limited values. For example, populations cannot grow indefinitely since resources such as food, water, and space are limited, so a logistic model best describes populations.

What is a carrying capacity? What kind of model has a carrying capacity built into its formula? Why does this make sense?

What is regression analysis? Describe the process of performing regression analysis on a graphing utility.

Regression analysis is the process of finding an equation that best fits a given set of data points. To perform a regression analysis on a graphing utility, first list the given points using the STAT then EDIT menu. Next graph the scatter plot using the STAT PLOT feature. The shape of the data points on the scatter graph can help determine which regression feature to use. Once this is determined, select the appropriate regression analysis command from the STAT then CALC menu.

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
More example of algebra and trigo
What is Indices
If one side only of a triangle is given is it possible to solve for the unkown two sides?
cool
Rubben
kya
Khushnama
please I need help in maths
Okey tell me, what's your problem is?
Navin