Section 8-2 Properties of Exponential Functions. Asymptote Is a line that a graph approaches as x or y increases in absolute value.

Slides:

Advertisements

Similar presentations

A logarithm with a base of e is called a natural logarithm and is abbreviated as “ln” (rather than as log e ). Natural logarithms have the same properties.

Advertisements

CONTINUOUSLY COMPOUNDED INTEREST FORMULA amount at the end Principal (amount at start) annual interest rate (as a decimal) time (in years)

Compound interest & exponential growth/decay. Compound Interest A=P(1 + r ) nt n P - Initial principal r – annual rate expressed as a decimal n – compounded.

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Warm Up Simplify. x 1. log 10 x 2. log b b 3w log z 3w3w z 4. b log b (x – 1 ) x – 1.

Chapter The Natural base, e.

8.2 Day 2 Compound Interest if compounding occurs in different intervals. A = P ( 1 + r/n) nt Examples of Intervals: Annually, Bi-Annually, Quarterly,

© 2008 Pearson Addison-Wesley. All rights reserved Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models.

Exponential Functions An exponential function is a function of the form the real constant a is called the base, and the independent variable x may assume.

1. Given the function f(x) = 3e x :  a. Fill in the following table of values:  b. Sketch the graph of the function.  c. Describe its domain, range,

1. 2 Switching From Exp and Log Forms Solving Log Equations Properties of Logarithms Solving Exp Equations Growth and Decay Problems

Chapter 8 Test #2 Review. Write In Exponential Form.

Pg. 255/268 Homework Pg. 277#32 – 40 all Pg. 292#1 – 8, 13 – 19 odd #6 left 2, up 4#14Graph #24 x = #28x = 6 #35 Graph#51r = 6.35, h = 9, V = 380 #1 Graph#3a)

Exponential Growth Exponential Decay Graph the exponential function given by Example Graph the exponential function given by Solution x y, or f(x)

Section 4.1 Logarithms and their Properties. Suppose you have $100 in an account paying 5% compounded annually. –Create an equation for the balance B.

Section 6.4 Solving Logarithmic and Exponential Equations

8.2 – Properties of Exponential Functions

Homework Questions.

Section 7.1: Graph Exponential Growth Functions Chapter 7: Exponential and Logarithmic Functions.

Compound Interest and Exponential Functions

Exponential Functions

Sect 8.1 To model exponential growth and decay Section 8.2 To use e as a base and to apply the continuously and compounded interest formulas.

Journal: Write an exponential growth equation using the natural base with a horizontal asymptote of y=-2.

Applications of Logs and Exponentials Section 3-4.

Section 4.2 Logarithms and Exponential Models. The half-life of a substance is the amount of time it takes for a decreasing exponential function to decay.

Algebra I Unit 7 Review. Unit 7 Review 1)A population of bacteria triples in size every day. a) Model the bacteria population with an exponential function.

Exponential Functions and Their Graphs Digital Lesson.

Simplify. 1. log10x 2. logbb3w 3. 10log z 4. blogb(x –1) 5.

– The Number e and the Function e x Objectives: You should be able to… 1. Use compound interest formulas to solve real-life problems.

3.1 (part 2) Compound Interest & e Functions I.. Compound Interest: A = P ( 1 + r / n ) nt A = Account balance after time has passed. P = Principal: $

9x – 7i > 3(3x – 7u) 9x – 7i > 9x – 21u – 7i > – 21u

GPS: MM3A2e, MM3A2f, MM3A2d.  MM3A2e – Investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes,

Homework Questions!.

Half – Life All radioisotopes have a half life. The half is of any given radioisotope is the amount of time required for one half (50%) of the sample to.

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

Chapter 8 Section 2 Properties of Exp. Functions

GRAPHING EXPONENTIAL FUNCTIONS f(x) = 2 x 2 > 1 exponential growth 2 24–2 4 6 –4 y x Notice the asymptote: y = 0 Domain: All real, Range: y > 0.

8.2 Properties of Exponential Functions An education isn't how much you have committed to memory, or even how much you know. It's being able to differentiate.

8.2 Properties of Exponential Functions 8.3 Logarithmic Functions as Inverses.

Pg. 255/268 Homework Pg. 277#32 – 40 all Pg. 310#1, 2, 7, 41 – 48 #6 left 2, up 4#14Graph #24 x = #28x = 6 #35 Graph#51r = 6.35, h = 9, V = 380 #1 Graph#3a)

7.2 Properties of Exponential Functions

Objectives Use the number e to write and graph exponential functions representing real-world situations. Solve equations and problems involving e or natural.

4.3 Use Functions Involving e PROJECT DUE: Tomorrow Quiz: Tomorrow Performance Exam: Friday *We will be having a book check tomorrow…. BRING BOTH.

Section 6: The Natural Base, e. U se the number e to write and graph exponential functions representing real-world situations. S olve equations and problems.

Holt McDougal Algebra The Natural Base, e Warm Up Simplify. x 1. log 10 x 2. log b b 3w log z 3w3w z 4. b log b (x – 1 ) x – 1 Slide 1 of 16.

Calculus Sections 5.1 Apply exponential functions An exponential function takes the form y = a∙b x where b is the base and b>0 and b≠1. Identify as exponential.

Drill If a quantity increases by the same proportion r in each unit of time, then the quantity displays exponential growth and can be modeled by the.

7-6 The Natural Base, e Entry Task Lesson 7.6 Practice Holt Algebra 2.

E XPONENTIAL W ORD P ROBLEMS Unit 3 Day 5. D O -N OW.

Obj: Evaluate and graph exponential functions. Use compound formulas. Warm up 1.Find the limit. x ,00050,000100,000150,000 y.

Warm Up:. 6.2 Notes: The Natural Base “e” The Basics  The natural base’s symbol is “e,” and is an irrational number (similar to pi). It is approximately.

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Recall the compound interest formula A = P(1 + )nt, where A is the amount, P is the principal, r is the annual interest, n is the number of times the.

The Natural Base, e Warm Up Lesson Presentation Lesson Quiz

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Chapter 5: Inverse, Exponential, and Logarithmic Functions

Algebra I Chapter 8 Review

6.1 Exponential Growth and Decay Functions

Algebra II H/G Section-07-02

Exponential Growth and Decay

Section 5.1 – Exponential Functions

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

The Natural Base, e 4-6 Warm Up Lesson Presentation Lesson Quiz

Choose the graph of the function y = 2 x from the following:

6.1 Exponential Growth and Decay Functions

Algebra II H/G Section-07-02

Algebra 2 Ch.8 Notes Page 56 P Properties of Exponential Functions.

Warm Up L8-2 Obj: Identify the role of constants in y = abcx.

The Natural Base, e 7-6 Warm Up Lesson Presentation Lesson Quiz

Presentation transcript:

Section 8-2 Properties of Exponential Functions

Asymptote Is a line that a graph approaches as x or y increases in absolute value.

Example Graph y = 3 2 x and y = –3 2 x. Label the asymptote of each graph.

Example Graph y = 3 2 x and y = –3 2 x. Label the asymptote of each graph.

Example 2 – Look at the graph and identify the asymptotes

Half - life Of a radio active substance is the time it takes for half of the material to decay or decompose. The decay factor is.5

Example Suppose a hospital prepares a 100-mg supply of technetium – 99m, which has a half life of 6 hours. Make a table showing the amount of technetium –99m remaining at the end of each 6- hour interval for 36 hours? Write the equation to describe the exponential function? Use the function to find the amount after 75 hours? # of 6hr half lives Technetium- 99m Present (mg)

Example Suppose a hospital prepares a 100-mg supply of technetium – 99m, which has a half life of 6 hours. Make a table showing the amount of technetium –99m remaining at the end of each 6- hour interval for 36 hours? Write the equation to describe the exponential function? Use the function to find the amount after 75 hours? # of 6hr half lives Technetium- 99m Present (mg)

Example Suppose a hospital prepares a 100-mg supply of technetium – 99m, which has a half life of 6 hours. Make a table showing the amount of technetium –99m remaining at the end of each 6- hour interval for 36 hours? Write the equation to describe the exponential function? Use the function to find the amount after 75 hours? # of 6hr half lives Technetium- 99m Present (mg)

Example Suppose a hospital prepares a 100-mg supply of technetium – 99m, which has a half life of 6 hours. Make a table showing the amount of technetium –99m remaining at the end of each 6- hour interval for 36 hours? Write the equation to describe the exponential function? Use the function to find the amount after 75 hours? # of 6hr half lives Technetium- 99m Present (mg)

Example Suppose a hospital prepares a 100-mg supply of technetium – 99m, which has a half life of 6 hours. Make a table showing the amount of technetium –99m remaining at the end of each 6- hour interval for 36 hours? Write the equation to describe the exponential function? Use the function to find the amount after 75 hours? # of 6hr half lives Technetium- 99m Present (mg)

The number e When interest is compounded continuously the formula can be simplified using the number e. The formula for continuously compounded interest is: A = P e rt

The number e When interest is compounded continuously the formula can be simplified using the number e. The formula for continuously compounded interest is: A = P e rt

Example You invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have after 5 years?

Example You invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have after 5 years?

Example You invest $1050 at an annual interest rate of 5.5% compounded continuously. How much will you have after 5 years?

Checking For Understanding Do not 1. Use the formula A = Pe rt to find the amount in a continuously compounded account where the principal is $2000 at an annual interest rate of 5% for 3 years. 2. Evaluate e to four decimal places. 1313