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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.

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• 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?
• Use parentheses in the exponent. 64*0.8^(t/1.5)
• 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.
• 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?
• Is there anyway of forming the function using the percentage that the population declined as apposed to the remaining population?
• If a population is decreasing by r% then the formula would be f(t)=P(1-r)^t
• Why doesn't this formula follow the geometric sequence formula Tn = ar^n-1? It does except for subtracting one from the exponent.
• 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.
• 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.
• Put parentheses around the exponent
for example: 2^(8/t)
• Why is the exponent what it is? Like why do we have to divide it by 2.8?
• 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!
• 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.
• 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!