3-1: Exponential Functions (day 1)

An exponential function with base b has the form f(x) = abx, where x is any real number and a andb are real number constants such that a ≠ 0, b is positive, and b ≠ 1. If b > 1, then the function is exponential growth. If 0 < b < 1, then the function is exponential decay.

Asymptote: A vertical (x=) or horizontal (y=) line that the graph approaches but never actually touches.

Increasing and Decreasing: We say a function is decreasing if the graph goes down as we move from left to right on the x-axis. We say a function is increasing if the graph goes up as we move from left to right on the x-axis.

New notation for end behaviors:

Example1). Sketch and analyze the graph of . Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

x / -3 / -2 / -1 / 0 / 1 / 2 / 3
f(x)
Domain: / Range: /
Intercept: / Asymptote:
End behavior:
Increasing and/orDecreasing:

Partner Work:Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing.

  1. h(x) = 2x − 1 + 1
  2. k(x) = e-2x
  3. f(x) = 2x – 1

Homework: Page 166: 1-10

3-1: Exponential Functions(day 2)

Exponential Growth and Decay

Many real-world situations can be modeled by exponential functions. One of the equations below may apply.

Exponential Growth or
Decay
N = N0(1 + r)t
N is the final amount, N0 is the initial amount, r is the rate of growth or decay, and t is time. / Continuous Exponential
Growth or Decay
N = N0ekt
N is the final amount, N0 is the initial amount, k is the rate of growth or decay, t is time, and e is a constant. / Compound Interest

P is the principal or initial investment, A is the final amount of the investment, r is the annual interest rate, n is the number of times interest is compounded each year, and t is the number of years.

Example 1 BIOLOGY A researcher estimates that the initial population of a colony of cells is 100. If the cells reproduce at a rate of 25% per week, what is the expected population of the colony in six weeks?

N = N0(1 + r)t
= 100(1 + 0.25)6
≈ 381.4697266 / Exponential Growth Formula
N0 = 100, r = 0.25, t = 6
Use a calculator.

There will be about 381 cells in the colony in 6 weeks.

Example 2 FINANCIAL LITERACY Lance has a bank account that will allow him to invest $1000 at a 5% interest rate compounded continuously. If there are no other deposits or withdrawals, what will Lance's account balance be after 10 years?

Partner Work

  1. FINANCIAL LITERACY Compare the balance after 10 years of a $5000 investment earning 8.5% interest compounded continuously to the same investment compounded quarterly.
  2. ENERGY In 2007, it is estimated that the United States used about 101,000 quadrillion thermal units. If U.S. energy consumption increases at a rate of about 0.5% annually, what amount of energy will the United States use in 2020?
  3. BIOLOGY The number of rabbits in a field showed an increase of 10% each month over the last year. If there were 10 rabbits at this time last year, how many rabbits are in the field now?

Homework: See worksheet.

NAME ______DATE ______PERIOD ______

(3-1) Exponential Functions Day 2 Homework.

  1. FINANCIAL LITERACY Suppose Jamal has a savings account with a balance of $1400 at a 4% interest rate compounded monthly. If there are no other deposits or withdrawals, what will be Jamal's account balance in three years?
  2. BIOLOGY Suppose a certain type of bacteria reproduces according to the model P(t) = 100e0.271t, where t is time in hours and P(t) is the number of bacteria.
  3. Determine the growth rate.
  4. What was the initial number of bacteria?
  5. Find the number of bacteria in 5, 10, 24, and 72 hours. Round to the nearest whole number.
  1. FINANCAL LITERACY You have $1000 to put into the bank. One bank offers a 5.7% interest rate compounded monthly. Another bank offers 5.6% compounded continuously. Which would you choose to make the most money after 2 years? after 5 years? Explain.
  2. TECHNOLOGY In 1965, Gordon Moore stated that since the invention of the integrated circuit in 1958, the number of transistors that can be placed on that circuit has doubled every two years. This statement has been true to the present day. Almost all measures of computing power come from this statement so that we can say that the computing power doubles nearly every two years.
  3. If there were about 2100 transistors on every circuit in 1971, write an exponential equation to model the number of transistors in a given year t after 1971.
  4. Approximately how many transistors were on one circuit in 2009?
  5. A 1971 computer could manage one process per second. Every two years, the number of processes also doubles on a computer. Write an equation to calculate the number of processes a computer can manage every second in each year after 1971. Then complete the table below.

1991 / 2001 / 2011 / 2021 / 2031
  1. If your precalculus teacher offers to give you 1 second of homework for the first week of school and double the amount of homework each week until the end of the school year (i.e. 2 seconds the second week), should you say yes? Explain.