Standard
A-SSE.4
A-CED.1
A-REI.11
Description
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the
formula to solve problems. For example, calculate mortgage payments.
Create equations and inequalities in one variable and use them to solve problems. Include equations
arising from linear and quadratic functions, and simple rational and exponential functions
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect
are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to
graph the functions, make tables of values, or find successive approximations. Include cases where f(x)
and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
F-IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and
using technology for more complicated cases.
1 Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric
functions, showing period, midline, and amplitude.
F-BF.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model
situations, and translate between the two forms.
Bring in formal recursive function notation.
F-BF.4
Find inverse functions.
1 Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression
for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.
F-LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity
increasing linearly, quadratically, or (more generally) as a polynomial function.
F-LE.4
For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and
the base b is 2, 10, or e; evaluate the logarithm using technology.
Students will be able to:
 Graph an exponential function showing the intercepts and end behaviors
 Graph a log function showing the intercepts and end behaviors
 Compare rates of growth for different functions (linear, quadratic, exponential, and polynomial) by using tables
and graphs
 Determine if a function has an inverse using the horizontal line test
 Find the inverse of a function if it exists
 If an inverse exists, use the inverse function to solve an equation of the form f(x)=c using equations, graphs and
tables. (clarification: find f-1(x) and then find f-1(c)….this will give the x-value that gives the solution of c of the
original equation)
 express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e.
 express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b>;0.
 Use technology to evaluate a logarithm
 Graph exponential functions symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases
 Graph logarithmic functions symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases
 Use the properties of logs to solve exponential and log equations
 Solve a system of equations that includes an exponential and/or logarithmic function(s) using technology
 Identify if a sequence is arithmetic or geometric or neither
 Derive the formula for the sum of a finite geometric series
 use the formula of a geometric series to solve problems
 Determine the common ratio for a geometric sequence.
 Write a geometric sequence given a rule from both recursive and explicit forms
 Use the formal recursive notation
 Determine a rule for a geometric sequence both recursively and explicitly
 Find a partial sum for a geometric sequence.
 Find an infinite sum for a geometric sequence.
 Use sigma notation for geometric series
Make the assumption that:
 There are 3,000 fish currently in the pond.
 1,000 fish are added at the end of each year.
 The population decreases by 20% each year (taking into account the combined effect of all
causes, including births, natural deaths, and fish being caught).
1. Using these assumptions, build a mathematical model to analyze the population growth in the pon
as follows.
a. Estimate of the population after one year. Estimate the population after two years.
Describe how you computed these estimates. What additional details did you assume to
get the answers that you have?
b. Assume that the population decreases by 20% before the 1,000 new fish are added. Also
assume that the population after each year is the population after the 1,000 new fish are
added. Are these the assumptions you used to compute your answers. In Part A? if not, go
back and recomputed using these assumptions. Explain why these assumptions are
reasonable. These are the assumptions you will use for the rest of this analysis.
c. Write a formula using the words NOW and NEXT to model this situation as specified in Part
b.
d. Use the formula from Part c and the last answer feature of your calculator or computer
software to find the population after seven years. Explain how the keystrokes or software
features you used correspond to the words NOW & NEXT in the formula.
2. Now think about the patterns of change in the long-term population of fish in the pond.
a. Do you think the population will grow without bound? Level off? Die out? Make a quick
guess about the long-term population.
b. Determine the long term population by continuing the work that you started in Part d of
Problem 1.
c. Explain why the long-term population ou have determined is reasonable. Give a general
explanation in terms of the fishing pond ecology. Also based on the assumptions above,
explain mathematically why the long-term population is reasonable.
d. Does the fish population change faster around year 5 or around year 25? How can you tell?
3. What do you think will happen to the long-term population of fish if the initial population is
different but all other conditions remain the same? Make an educated guess. Then check your
guess by finding the long-term population for a variety of initial populations. Describe the pattern
of change in long-term pooulation as the initial population varies.
4. Investigate what happens ot the long-term population if the annual restocking amount changes
but all other conditions are thee same as in the original assumptions. Describe as completely as
you can the relationship between long-term population and restocking amount.
5. Describe what happens to the long-term population if the annual decrease rate changes but all
other conditions re the same as in the original assumptions. Describe the relationship between
long-term population and the annual decrease rate.
6. Now consider a situation in which the fish population shows an annual rate of increase.
a. What do you think will happen to the long-term population if the population increases at a
constant annual rate? Make a conjecture and then teset it by trying at least two different
annual increase rates.
b. Write formulas using NOW and NEXT that represent your two test cases.
c. Do you think it is reasonable to model the population of fish in a pond with an annual rate
of increase? Why or why not?
7. Graphs provided below
This labels the population of the fish in the next 50 years. Following
the assumptions, and from looking at the data, it is clear that the
population of fish will never exceed 5,000. Since 20% of the
population of fish is expected to perish each year, 20% of 5,000 is
equal to 1,000. That is the number of fish that becomes put back into
the lake. Therefore we are putting refilling the same number that is
disappearing. If we were to graph this data it would look like this.
This is called a geometric sequence. We can find the limit of a
𝐴
geometric sequence by: where A is the initial value and r is the rate
𝑟
of change. Notice that 3,000 was not the initial value because even if
the starting population was 0, the limit would still be 5,000.