8.1 Growth and Decay
Name__________________________________
I. The Game of Chess(Open the TI-NSpire file, Exponential Functions.tns)
1. Without doing any calculations, do you think the king’s promise was wise?
2. Complete the table for the first 6 squares.
Square
Number of Grains
Exponential Form
1
1
-2
2
2
-2
3
4
-2
4
5
6
n
X
64
3. Write an expression for the number of grains on square n of the chess board.
4. A chessboard has 64 squares. How many grains of rice are to be placed on the 64th square?
5. To get an idea how big this number is, suppose that there are 1,000,000 people in the kingdom and that each grain
of rice weighs 0.00008 pounds. How many pounds of rice per person does the rice on the 64th square represent?
6. Do you think the king was able to fulfill the request?
II. Cell Phone Subscribers: The table gives the number y (in millions) of cell phone subscribers from 1988 to 1997,
where t is the number of years since 1987.
t
y
1
1.6
2
2.7
3
4.4
4
6.4
5
8.9
6
13.1
7
19.3
8
28.2
9
38.2
10
48.7
1. Draw a scatterplot of subscribers versus time.
2. What is the basic shape of the data?
3. Find a model (equation) that fits the data.
4. Use your model to predict the number of cell phone subscribers this year .
5. What is the percentage of increase/decrease of the model?
III. Paper Folding
What relationships exist between exponential growth and decay when a piece of paper is folded repeatedly?
1. Fold a rectangular piece of paper in half. The fold divides the paper into two regions, each of which has half the
area of the paper.
2. Fold the paper in half again. Into how many regions has the original piece of paper been
folded? What fraction of the paper's area does each region have?
3. Continue to fold the paper until it is no longer possible to make another fold. Record the
fold number, the number of regions into which the paper has been folded, and the
fraction of the paper's area that each region has.
Fold Number
0 1
2 3 4
Number of Regions
1 2
Fractional Area of Each Region 1 1
2
4. Graph (fold number, number of regions) on the first scatterplot. Is it an example of
exponential growth or decay? Write an equation for the graph.
5. Use the equation to determine the number of regions after 8 folds.
6. Graph (fold number, fractional area of each region) on the second scatterplot. Is it an
example of exponential growth or decay? Write an equation for the graph.
7. Use the equation to determine the fractional area of each region after 8 folds.
IV. Graphing Exponential Functions.
1. Explore several different b-values by dragging the slider.
a. For what b-values is the function increasing? Why?
2.
3.
b.
For what b-values is the function decreasing? Why?
c.
For each b -value, identify the y-intercept of the function. Interpret your results.
d.
When b > 0, why is there no x-intercept?
Drag the slider to explore several different b-values where b > 1.
a. What is the domain of the function?
b.
What is the equation of the horizontal asymptote?
c.
What is the range of the function?
Drag the slider to explore several different b-values where 0 < b < 1.
a. Describe how the graph changes?
b.
Does this change the domain, horizontal asymptote, or range?
( )
Exponential decay functions are of the form f ( x ) = a ⋅ bx , where a > 0
Exponential growth equations are of the form f x = a ⋅ bx , where a > 0 and b > 1 .
Exponential functions have a horizontal asymptote of
V. Practice
x
Ex. y = 2
y-intercept:_______
growth or decay:_______
domain:_______
asymptote:_______
range:_______
1.
and 0 < b < 1 .
y = 0 unless they have a vertical shift.
y = 2x + 4
2.
y = −2 x − 3
3. y = 2
( x+3)
y-intercept:_______
y-intercept:_______
y-intercept:_______
growth or decay:_______
growth or decay:_______
growth or decay:_______
domain:_______
domain:_______
domain:_______
asymptote:_______
asymptote:_______
asymptote:_______
range:_______
range:_______
range:_______
4. The parent graph 𝑓 𝑥 = 3 ! translated left 4 units and up 5. The function for the stated translation is?
5. The parent graph 𝑓 𝑥 = −5 ! translated right 6 units and down 1. The function for the stated translation is?
6. The parent graph 𝑓 𝑥 = 8 ! reflected across the x-axis and translated right 3 units. The function for the stated
translation is?
7.
y = 1.5(1.42)x
8. y =
4(.37)x
9. y =
7(3)x
y-intercept:_______
y-intercept:______
y-intercept:_______
growth or decay:_______
growth or decay:_______
growth or decay:_______
growth/decay factor:_______
growth/decay factor:_______
growth/decay factor:_______
% increase or decrease:_______
% increase or decrease:_______
% increase or decrease:_______
10. Write an exponential function with a starting point of 12 and a growth rate of 54%.
11. Write an exponential function with a starting amount of 200 and a decay rate of 54%.