Algebra II Chapter 8 Section 1.
Exploring Exponential Models
1) Make a table from -3 to 3 for the equation y = 2x
x
-3
-2
-1
0
1
2
3
x
2
y
a) Graph the points on the axis to the right.
b) What is any number raised to the zero power?
c) What is the y –intercept for y = 3(4)x ? (Hint: what is the value of x when the graph crosses the
x-axis?)
So…You have just seen the basic form for an exponential function.
y = abx
Where the control variable, x, is an exponent (hence the term, exponential function!! )
2) In your calculator, graph y = 20(1.15)x versus y = 20(0.85)x
(Use the window: -20 < x < 20, count by 2; 0 < x < 20, count by 1)
a) What is the y-intercept for both equations?
b) What is the growth rate (b) in each equation?
c) Make a generalization for what a graph does when.
b>1
and
b<1
3) If the population of the United States was 281 million people in 2000 and grows at a rate of 1.14%
a) Write a function that best models the population for any year after 2000.
b) If x represents years after 2000, what will the population be in 2008?
c) Use your model to predict the population in 1950.
3) 20 tigers are dumped on a deserted island and told to get busy. If they reproduce exponentially at
a rate of 2.5%
a) Write an exponential model for this situation.
b) What will the population be 10 years from now?
25 years from now?
d) Use the table in your calculator to figure out when the population will reach 450 tigers.
4) Write an exponential function that includes the two points: (2, 2) and (3, 4)
5) The table below represents degrees above room temperature for a cup of coffee after x minutes of
cooling.
Time 0
5
10 15 20 25 30 35 40 45 50 55 60
Temp 135 100 74 55 41 30 22 17 12 9 7 5 4
a) Use STAT PLOT to draw these points in your calculator.
b) Run an exponential regression for this data.
c) Graph this line on your model. How does it fit?
The line of best fit can be modeled by
f(x) =
d) Use the table function on your calculator. At what time is the temperature of the coffee 65
degrees?
6) The NCAA basketball tournament starts with 64 teams (after the play in game) and eventually
crowns a college basketball champion. Complete the table.
Create a table for teams left after round x of the tournament. The table is complete when there is only
one team remaining.
After round x
Number of teams
left in tournament y
a) How many rounds are played?
b) How does the number of teams left in each round compare to the number of teams in the
previous round?
c) Write an exponential model for this (what is the rate of decay?). Plug in some values from
your table to make sure it works.
7) You bought a car for 20,000. One year after the purchase the value of the car is 17,000.
a) What percentage of the car did you lose?
b) What percent of the value of the car do you retain each year? (How much did you have LEFT
each year?) This is the growth factor, b
c) Put this info into an exponential model.
d) Find the value of the car in 6 years.
Check answers for Problem 7 on pg. 434 and complete #9, 10, 13, 16, 17, 19, 20, 32, 35, 43, 45