Unit 3 - 5.1 Exponential Functions
Our goals for today’s lesson:






(1)
(2)
(3)
(4)
(5)
(6)
Write explicit equations for geometric sequences
Define and write exponential functions as y = abx.
Solve real-world growth and decay problems and how they relate to b.
Learn about half-life and doubling time
Evaluate exponential functions using explicit equations or graphical methods
Discover point-ratio form.
-------------------------------------------------------------------------------------------------------------------------------------------------------------Vocabulary
Geometric Growth
Geometric Decay
(Goal2) Exponential function
(Goal4) Half-life
(Gaol4) Doubling time
---------------------------------------------------------------------------------------------------------------------------------------------------------------Example A: Most automobiles depreciate as they get older. Suppose an automobile that originally costs $14,000
depreciates by one-fifth of its value every year. (Goal2, 3, 4, 5)
a. What is the value of this automobile after two and a half years?
b. When is this automobile worth half of its initial value?
Example B: The functions g(x) =
1 x
2 and h(x) = 2x-1 are transformations of the parent function f(x) = 2x. Describe the
2
transformations and sketch the graphs. What do you notice about g(x) and h(x)?
Investigation Radioactive Decay
How we are going to collect our data:
1)
All members of the class should stand up, except for the recorder. The recorder counts and records the number
standing at each stage.
2) Each standing person rolls a die, and anyone who gets a 1 sits down.
3) Wait for the recorder to count and record the number of people standing.
4) Repeat Steps 2 and 3 until fewer than three students are standing.
---------------------------------------------------------------------------------------------------------------------------------------------------------------Step 1: Record the number of students in the second column of the table below.
Step 2: Graph your data in your calculator. Is the graph linear? What type of sequence does this resemble?
n
Un
Un in terms of u0 and r
Un in terms of u0 and r
using exponents
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Step 3: How can we find the approximate common ratio of our sequence?
Step 4: Use u0 and r to complete the table above to write an explicit formula for our data. (Goal1)
Step 5: Graph the explicit formula on your calculator. Notice where the value of u0 appears in your equation. Your
graph should pass through the original data point (0, uo).
Step 6: Modify your equation so that it passes through (1, u1), the second data point. (Think about translating the graph
horizontally and also changing the starting value.)
Step 7: Experiment with changing your equation to pass through other data points. Decide on an equation that you
think is the best fit for your data. Write a sentence or two explaining why you chose this equation.
Step 8: What equation with ratio r would you write that contains the point (6, u6)? (Goal6)
Apply What You Learned and Practice
1.
Evaluate each function at the given value.
a. f(x) = 4.753(0.9421)x, x = 5
c.
2.
h(t) = 47.3(0.835)t + 22.3, t = 24
b. g(h) = 238(1..37)h, h = 14
d. j(x) = 225(1.0825)x-3, x = 37
Record three terms of the sequence and then write an explicit function for the sequence.
a. u0 = 16
b. u0 = 24
un = 0.75un-1
un = 1.5un-1
3.
Evaluate each function at x = 0, x = 1, and x = 2 and then write a recursive formula for the pattern.
a. f(x) = 125(0.6)x
b. f(x) = 3(2)x
4.
Calculate the ratio of the second term to the first term, and express the answer as a decimal value. State the
percent increase or decrease.
a. 48, 36
b. 54, 72
c. 50, 47
d. 47, 50
5.
In 1995, the population of the People’s Republic of China was approximately 1.211 billion, with an annual
growth rate of 1.5%.
a. Write a recursive formula that models this growth. Let u0 represent the population in 1995.
b. Complete a table recording the population for years 1995 to 2002.
Year
1995
Population
1996
1997
1998
1999
2000
2001
2002
c.
Define the variables and write an exponential equation that models this growth. Choose two data points
from the table and show that your equation works.
d.
One estimate for the population of China in 2006 was 1.314 billion. How does this compare with the value
predicted by your equation? What does this tell you?
6.
Jack planted a mysterious bean just outside his kitchen window. It immediately sprouted 2.56 cm above the
ground. Jack kept a careful log of the plant’s growth. He measured the height of the plant each day at 8:00 A.M.
and recorded these data:
Day
0
1
2
3
4
Height (cm)
2.56
6.4
16
40
100
a.
Define variables and write an exponential equation for this pattern. If the pattern continues, what will be the
heights on the fifth and sixth days
b.
Jack’s younger brother measured the plant at 8:00 p.m. on the evening of the third day and found it to be about
63.25 cm tall. Show how to find this value mathematically. (You may need to experiment with your calculator).
c.
Find the height of the sprout at 12:00 noon on the sixth day.
d. Find the doubling time for this plant.
e. Experiment with the equation to find the day and time (to the nearest hour) when the plant reaches a height of
1 km.
7. Each of the black curves is a transformation of the graph of y = 0.5x, shown as the dashed line. Focus on how the two
marked points on the dashed curve are transformed to become the corresponding points on the black curve. Write an
equation for each black curve.
a.
b.
c.
d.
8. Austin deposits $5,000 into an account that pays 3.5% annual interest. Sami deposits $5,200 into an account that
pays 3.2% annual interest.
a. Write an expression for the amount of money Austin will have in his account after 5 years if he doesn’t deposit any
more money.
b. Write an expression for the amount of money Sami will have in her account after 5 years if she doesn’t deposit any
more money.
c. How long will it take until Austin has more money than Sami?
Name: ____________________________________
Algebra II
Homework 5.1 – Exponential Functions
Unit 3
Use your notes, investigation, and the Big Ideas from the Chapter 5 Overview to help you.
1. Evaluate each function at the given value. Round to four decimal places, if necessary.
a. r(t) = 325(1 + 0.035)t, t = 8
b. f(x) = 59.5(1 – 0.095)x; x = 10.
2. Record the next three terms for each sequence. Then write an explicit function for the sequence.
a. a0 = 12
b. u0 = 50.5
an = 0.8an-1
un = 2.1un-1
Terms: __________________
Terms: __________________
Explicit formula: _________________________
Explicit formula: _________________________
3.
Evaluate each function a x = 0, x = 1, and x = 2 and write a recursive formula for the pattern. Then, indicate
whether each equation is a model for exponential growth or decay.
a. f(x) = 2000(0.9)x
b. f(x) = 3000(1 + 0.001)x
c. f(x) = 0.1(1 – 0.5)x
x=0
____________
x=0
____________
x=0
____________
x= 1
____________
x= 1
____________
x= 1
____________
x=2
____________
x=2
____________
x=2
____________
Recursive _______________________ Recursive _______________________ Recursive _______________________
Growth/Decay ___________________ Growth/Decay ___________________ Growth/Decay ___________________
4.
Calculate the ratio of the second term to the first term, and express the answer as a decimal value. State the
percent increase or decrease.
a. 80, 60
b. 36, 32
c. 63, 100.8
Ratio: ________________
Ratio: ________________
Ratio: ________________
Percent change: __________________ Percent change: __________________ Percent change: __________________
5.
Rohit bought a new car for $17,500. The value of the car depreciating at a rate of 16% a year.
a. Write a recursive formula that models this situation. Let u0 represent the purchase price, u1 represent the
value of the car after 1 year, and so on.
Recursive formula ________________________________________
b.
Define variables and write an exponential equation that models this situation.
Variables ______________________________
___________________________________
Equation _________________________________________________
8. Each of the black curves is a transformation of the graph of y = 2x, shown as the dashed line. Focus on how the
two marked points on the dashed curve are transformed to become the corresponding points on the black curve.
Write an equation for each black curve.
a.
b.
Equation _________________________
Equation _________________________
f.
d.
Equation _________________________
Equation _________________________