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### Course: Algebra 2 > Unit 7

Lesson 2: Constructing exponential models according to rate of change# Constructing exponential models: percent change

Sal models a population of narwhals using an exponential function.

## Want to join the conversation?

- On the Next Exercise called "Construct Exponential Models" I am not able to make the fraction t/etc because the machine stops showing it as an exponent and it messes with the equation

example: 64*0.8^t/1.5 but the fraction in the exponent does this

64*(0.8^t)/1.5

How do I fix this?

Please Help(2 votes)- Use parentheses in the exponent. 64*0.8^(t/1.5)(21 votes)

- Are there any lessons on rewriting these functions so they are simplified to y=ab^x? My teacher does not accept an answer written as, for example, y=40(8)^t/3. Such a function would have to simplified to y=40(2)^t. If there were some lessons or practice exercises on the topic of rewriting functions, I would find that extremely hepful.(3 votes)
- There probably aren't any videos on rewriting functions in the sense you mean. What you mean is just simplifying your answer, and that depends on your teacher's requirements. If you're required to simplify to ab^x, there isn't a way for one video to teach that because it overlaps with other topics.

Do you understand how 40(8)^t/3 was simplified into 40(2)^t?(7 votes)

- Is there anyway of forming the function using the percentage that the population declined as apposed to the remaining population?(4 votes)
- If a population is decreasing by r% then the formula would be f(t)=P(1-r)^t(5 votes)

- Why doesn't this formula follow the geometric sequence formula Tn = ar^n-1? It does except for subtracting one from the exponent.(2 votes)
- It depends on how the question is phrased. Notice how the exercise in the video says: "write a function that models the population of the narwhals t months since the beginning of the population".

They are asking you to give them a formula that gives the population t months from the**beginning**of the population. So t = 0, means 0 months from the beginning, and should give you the population at the beginning of the study, which is 89.000.

Notice that if we set t = 1, we are nog longer talking about the beginning. We are now talking about 1 month since the beginning of the study.

Now think about what kinds of questions we are answering when writing geometric series Formulas. Usually they will ask you to write a formula that will give you the full sequence of terms of a geometric serie. So this time t = 1, and this should give you the first term of the serie. That is why with a geometric series formule you subtract 1 from the exponent. But this al depends on how the question is phrased.

Ask yourself the following question: where t >= 0. Are they asking the the growth since the initial size as the output of a given function? Then t = 0 should give you the initial size.

Or are they asking to include the initial size as a possible output to the given formula? Then t = 1 should give you the initial size.

So in conclusion: it depends on how the questions for a given problem is phrased. Modeling exponential equations or geometric sequence equations, can both be written in different forms.(4 votes)

- i can enter an exponent using the on screen display, but I'm having problem entering a fractional exponent like 2.8/t. when I use the division slash or the two boxes on the display it creates a division for the entire expression. i cannot continue with the exercises until i know how to do this.(2 votes)
- Put parentheses around the exponent

for example: 2^(8/t)(3 votes)

- Why is the exponent what it is? Like why do we have to divide it by 2.8?(2 votes)
- The
**exponent**tells us how many times to multiply the initial amount by the base.

We are given that every`2.8 months`

, population*decreases*by`5.6%`

. If we use`t`

as time in months, it is intuitive that an increase by`2.8 months`

must result in`1`

more multiplication by the base; by`1 - 5.6% or 0.944`

. So, the exponent is`t months`

divided by the specified time, which gives`t months/2.8 months = t/2.8`

.

Hence why the exponential function is:`P(t) = 89,000 * 0.944^(t/2.8)`

.

In general, the exponent is set to the time variable of divided by the specified length of time for a growth/decay. Thereafter, handle the units.

For example if it's every`6`

months and you make a function in terms of`t`

years, then the exponent should be`t years/6 months = 12t months/6 months = 2t`

.

Hope this helped!(3 votes)

- how can I know when the population of narwhals will be 0, I imagine the solution to be (0.944)^m/2.8 = 0

i.e ((0.944)^1/2.8)^m = 0

= log base((0.944)^1/2.8)(0) = number of months for narwhals to be extinct,but I dont understand how this gives me undefined.(3 votes) - I have a question about the "Practice: Construct Exponential models" exercise. A certain population will be decreasing by a fractional amount every so many months. Sometimes the common ratio is the original fraction. Other times you subtract the fraction from one and use the remaining amount as the common ratio. How can I tell when and when not to use the original fraction as the common ratio?

Thank you!(3 votes) - but if t is -1? does it mean that the number of the narwhals 1 month before?(2 votes)
- Yes. It t=-1, then you would be referring to the month before the study started.(3 votes)

- How do you convert it to percentages?(2 votes)
- multiply the decimals by 100 :)(2 votes)

## Video transcript

- [Voiceover] Chepi is an
ecologist who studies the change in the narwhal population of
the Arctic ocean over time. She observed that the
population population loses 5.6% of its size every 2.8 months. The population of narwhals can be modeled by a function, N, which
depends on the amount of time, t in months. When Chepi began the study, she observed that there were 89,000
narwhals in the Arctic ocean. Write a function that models
the population of narwhals t months since the
beginning of Chepi's study. Like always, pause the video and see if you can do it on your own before we work through it together. So let's now work through it together. To get my essence of what
this function is to do, it's always valuable to
see, to create a table for some interesting
inputs for the function and seeing how the function should behave. So first of all, if t is in months and N of t is, in N is the, that models, N is the number
of narwhals, the narwhals. So what, when T is equal to zero, what is N of zero? Well, we know a T equals zero. There are 89,000 narwhals in the ocean. So 89,000. And now, what's another interesting one? Well, T is in months and we know that the population decreases 5.6% every 2.8 months. So let's think about when T is 2.8, 2.8 months. Well then the population, it should have gone down 5.6%. So going down 5.6% is the same thing as retaining. What's one minus 5.6%? Retaining 94.4%. I'll be clear. 100%, if you lose 5.6%, you are going to be left with 94.4%. The 0.6 plus 0.4 gets you a 95 plus another five is 100. So another way of saying, this sentence, that the population loses 5.6%
of its size every 2.8 months is to say that the population is 94%, 94.4% of its size every 2.8 months or shrinks to 94.4% of its original size every, or let me phrase this clearly. After every 2.8 months, the population, you can either say it shrinks 5.6% or you could say it has, it's gone from, it's 94.4% of the
population at the beginning of those 2.8 months. So after 2.8 months, the
population should be 89,000 times, I could write times 94.4% or I could write times 0.944. Now, if we go another 2.8 months, so two times 2.8. I have to say you could
just write that as, I could write that as 5.6 months but let me just write this is 2.8 months. Where are we gonna be? We're gonna be 89,000 times 0.944. This is where we were before at the beginning of this period. Now, we're gonna be 94.4% of that. So we're gonna multiply by 94.4% again or 0.944 again or we can just say times 0.944 squared. And after three of these periods, well, we're gonna be times 0.944 again. So it's gonna be 89,000 times 0.944 squared times 0.944 which is gonna be 0.944
to the third power. And I think you might
see what's going on here. We have an exponential function. Between every 2.8 months, we are multiplying by this common ratio of 0.944. And so we could write our function N of t. Our initial value is 89,000 times 0.944 to the power of, however, many of these 2.8 month period we've gone so far. So if we take the number of months and we divide by 2.8, that's how many 2.8 month period we have gotten gone. And so notice, when t equals zero, all of this turns into one, erasing the zero part, that becomes one. You have 89,000. When t is equal to 2.8,
this exponent is one. Now we're gonna multiply by 0.944 once. When t is 5.6, the exponent
is going to be two. Now we're gonna multiply by 0.944 twice. And I'm just doing the values that make the exponent integers
but it's going to work for the ones in between. I encourage you to graph it or to try those values in
a calculator if you like. But there you have it, we're done. We have modeled our narwhals. So let me just underline
that and we're done.