# 4.8 Fitting exponential models to data  (Page 7/12)

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What might a scatterplot of data points look like if it were best described by a logarithmic model?

What does the y -intercept on the graph of a logistic equation correspond to for a population modeled by that equation?

The y -intercept on the graph of a logistic equation corresponds to the initial population for the population model.

## Graphical

For the following exercises, match the given function of best fit with the appropriate scatterplot in [link] through [link] . Answer using the letter beneath the matching graph.

$y=10.209{e}^{-0.294x}$

$y=5.598-1.912\mathrm{ln}\left(x\right)$

C

$y=2.104{\left(1.479\right)}^{x}$

$y=4.607+2.733\mathrm{ln}\left(x\right)$

B

$y=\frac{14.005}{1+2.79{e}^{-0.812x}}$

## Numeric

To the nearest whole number, what is the initial value of a population modeled by the logistic equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{175}{1+6.995{e}^{-0.68t}}?\text{\hspace{0.17em}}$ What is the carrying capacity?

$P\left(0\right)=22\text{\hspace{0.17em}}$ ; 175

Rewrite the exponential model $\text{\hspace{0.17em}}A\left(t\right)=1550{\left(1.085\right)}^{x}\text{\hspace{0.17em}}$ as an equivalent model with base $\text{\hspace{0.17em}}e.\text{\hspace{0.17em}}$ Express the exponent to four significant digits.

A logarithmic model is given by the equation $\text{\hspace{0.17em}}h\left(p\right)=67.682-5.792\mathrm{ln}\left(p\right).\text{\hspace{0.17em}}$ To the nearest hundredth, for what value of $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ does $\text{\hspace{0.17em}}h\left(p\right)=62?$

$p\approx 2.67$

A logistic model is given by the equation $\text{\hspace{0.17em}}P\left(t\right)=\frac{90}{1+5{e}^{-0.42t}}.\text{\hspace{0.17em}}$ To the nearest hundredth, for what value of t does $\text{\hspace{0.17em}}P\left(t\right)=45?$

What is the y -intercept on the graph of the logistic model given in the previous exercise?

y -intercept: $\text{\hspace{0.17em}}\left(0,15\right)$

## Technology

For the following exercises, use this scenario: The population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of a koi pond over $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ months is modeled by the function $\text{\hspace{0.17em}}P\left(x\right)=\frac{68}{1+16{e}^{-0.28x}}.$

Graph the population model to show the population over a span of $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years.

What was the initial population of koi?

$4\text{\hspace{0.17em}}$ koi

How many koi will the pond have after one and a half years?

How many months will it take before there are $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ koi in the pond?

about $\text{\hspace{0.17em}}6.8\text{\hspace{0.17em}}$ months.

Use the intersect feature to approximate the number of months it will take before the population of the pond reaches half its carrying capacity.

For the following exercises, use this scenario: The population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ of an endangered species habitat for wolves is modeled by the function $\text{\hspace{0.17em}}P\left(x\right)=\frac{558}{1+54.8{e}^{-0.462x}},$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is given in years.

Graph the population model to show the population over a span of $\text{\hspace{0.17em}}10\text{\hspace{0.17em}}$ years.

What was the initial population of wolves transported to the habitat?

$10\text{\hspace{0.17em}}$ wolves

How many wolves will the habitat have after $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years?

How many years will it take before there are $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ wolves in the habitat?

Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

For the following exercises, refer to [link] .

 x f(x) 1 1125 2 1495 3 2310 4 3294 5 4650 6 6361

Use a graphing calculator to create a scatter diagram of the data.

Use the regression feature to find an exponential function that best fits the data in the table.

Write the exponential function as an exponential equation with base $\text{\hspace{0.17em}}e.$

$f\left(x\right)=776.682{e}^{0.3549x}$

Graph the exponential equation on the scatter diagram.

Use the intersect feature to find the value of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ for which $\text{\hspace{0.17em}}f\left(x\right)=4000.$

When $\text{\hspace{0.17em}}f\left(x\right)=4000,$ $x\approx 4.6.$

For the following exercises, refer to [link] .

 x f(x) 1 555 2 383 3 307 4 210 5 158 6 122

Use a graphing calculator to create a scatter diagram of the data.

can you not take the square root of a negative number
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
Suppose P= {-3,1,3} Q={-3,-2-1} and R= {-2,2,3}.what is the intersection
can I get some pretty basic questions
In what way does set notation relate to function notation
Ama
is precalculus needed to take caculus
It depends on what you already know. Just test yourself with some precalculus questions. If you find them easy, you're good to go.
Spiro
the solution doesn't seem right for this problem
what is the domain of f(x)=x-4/x^2-2x-15 then
x is different from -5&3
Seid
All real x except 5 and - 3
Spiro
***youtu.be/ESxOXfh2Poc
Loree
how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
By using some imaginary no.
Tanmay
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
0=3x-2 2=3x x=3/2 then . y=3/2X-2 I think
Given
co ordinates for x x=0,(-2,0) x=1,(1,1) x=2,(2,4)
neil
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Where do the rays point?
Spiro
x=-b+_Гb2-(4ac) ______________ 2a
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
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
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
Â
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