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Warm-up 7.1

Warm-up 7.1. What are the zeros of the function : y = x 3 – 4x 2 + x + 6? What is the discriminant of 2x 2 – 5x + 4=0? What is the end behavior of the function: f(x) = 5x 4 -3x 3 + x – 2?. Did You Know?.

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Warm-up 7.1

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  1. Warm-up 7.1 • What are the zeros of the function : y = x3 – 4x2 + x + 6? • What is the discriminant of 2x2 – 5x + 4=0? • What is the end behavior of the function: f(x) = 5x4 -3x3 + x – 2?

  2. Did You Know? Q. What do bulletproof vests, fire escapes, windshield wipers, and laser printers all have in common?A. All invented by women. F: One roach can live on a piece of gum for 5 years. Q. If you were to spell out numbers, how far would you have to go until you would find the letter "A"?A. One thousand • When a coffee seed is planted, it takes five years to yield it's first consumable fruit. • The common goldfish is the only animal that can see both infra-red and ultra-violet light.

  3. Tennessee is bordered by more states than any other. The eight states are Kentucky, Missouri, Arkansas, Mississippi, Alabama, Georgia, North Carolina and Virginia. • Des Moines has the highest per capita Jello consumption in the U.S • The Western-most point in the contiguous United States is Cape Alava, Washington. • There are only three animals with blue tongues, the Black Bear, the Chow Chow dog and the blue-tongued lizard.

  4. Grades, Papers, Folders

  5. §7.1: Exponential Models LEQ: How are exponential models similar to and different from linear and quadratic models? Introduction – Exponential Decay

  6. Exponential Function Y = abx “a” is a non-zero constant “b” is the base (greater than 0 and not equal to 1) “x” is any real number

  7. Exponential Growth Y = abx When b > 1, the function models exponential growth “b” is the growth factor

  8. Exponential Growth

  9. Exponential Decay y = abx When “b” is between 0 and 1 (is a positive fraction), the function models exponential decay. “b” is the decay factor

  10. Exponential Decay

  11. Identify each as exponential growth or decay. What is the percent increase/decrease? • Y = 1298(1.63)x • Y = 2(0.65)x • Y = 12(1.7)x • Y = 16(0.25)x • growth, 63% • Decay; 35% • Growth, 70% • Decay; 75%

  12. Real World The population of the United States in 1994 was about 260 million, with an average annual rate of increase of about 0.7%. FYI - Depreciation is the decline in an items value (exponential decay).

  13. What is the growth factor for the U.S. population? After one year, the population would be 260 + 260(0.7%) million. = 260 + 260(0.7%) = 260 + 260(0.007) = 260(1 + 0.007) = 260(1.007) The growth factor is 1.007

  14. Write an equation that models the future growth of the U.S. population. To write an equation, find “a” and “b”. X = # years after 1994 Y = the population (in millions) The population increases exponentially: Y = abx Y = 260(1.0007)x

  15. Use the equation to predict the population of the U.S. in 2001. Y = 260(1.0007)x X is the # of years since 1994, so x = 7 Plug it in… The population would be about 273 million.

  16. A handy “formula” • Percent increase/decrease refers to the “slope” of any given line, but typically asks for decrease/increase on a changeable slope (initial value – final value) Initial value

  17. Cars are an example of an item that depreciates exponentially. • Can you think of five other items that might depreciate exponentially?

  18. How do you determine whether an equation shows growth or decay? • How do you calculate the percent increase/decrease of an exponential growth equation? • How do you determine if a graphs shows exponential decay or growth?

  19. Assignment Section 7.1 p. 308 - 309 #5, 7, 9, 10, 11, 19 – 23 (10 problems total) Homework:

  20. Warm – up 10/22/08 Complete Technology Connection on pg. 311 in textbook. #1-4

  21. Did you know? • In 1886, Coca-cola was first served at a pharmacy in Atlanta, Georgia for only five cents a glass. A pharmacist named John Pemberton created the formula for Coca-cola • Flamingos are able to fly at a speed of approximately 55 kilometers an hour. In one night they can travel about 600 km • Men are able to read fine print better than women can • On average, 150 couples get married in Las Vegas each day • Spiders usually have eight eyes, but still they cannot see that well • In humans, the epidermal layer of skin, which consists of many layers of skin regenerates every 27 days

  22. Camel's milk does not curdle. • The ball on top of a flagpole is called the truck. • People generally read 25% slower from a computer screen compared to paper • Certain female species of spiders such as the Australian crab spider, sacrifice their bodies as a food source for their offspring • One grape vine produce can produce about 20 to 30 glasses of wine. • The Hubble telescope is so powerful that it is like pointing a beam of light at a dime that is two hundred miles away. • On average people fear spiders more than they do death • Every day, over five billion gallons of water are flushed down toilets in the United States • In one trip, a honey bee visits about 75 flowers • Jupiter is the fastest rotating planet, which can complete one revolution in less than ten hours • A chicken loses its feathers when it becomes stressed

  23. Review? • How do you determine whether an equation shows growth or decay? • How do you calculate the percent increase/decrease of an exponential growth equation? • How do you determine if a graphs shows exponential decay or growth?

  24. Assignment Section 7.1 p. 308 - 309 #5, 7, 9, 10, 11, 19 – 23 (10 problems total)

  25. Applications of exponential Equations…

  26. 7.2: Exponential Functions LEQ: What are the roles of the constants in the formula y = ab (kx) ? Compound Interest Formula A = P(1 + r/n)nt P = principal t = time in years R = rate of interest N = # of times per year interest is compounded

  27. Finding Compound Interest

  28. Ex. Write an equation to describe the exponential function of the form y = abx with base 2 whose graph passes through the point (3,4). 4 = a23 4 = a(8) ½ = a The equation would be : y = ½ 2 x

  29. You try: Write an equation to describe the exponential function y = abx with base 4 whose graph passes through the point (2,3). y = 3/16 4x

  30. Half-Life Half life of a radioactive substance is the time it takes for half of the material to decay or decompose (thus, it has a decay factor or 0.5) Half Life Formula y = abt/h a = initial amount b = 0.5 (half-life) t = time passed h = half life of item

  31. Suppose a hospital prepares a 100mg supply of technetium-99m which has a half-life of 6 hours. Find the amount after 75 hours. a = 100 (initial amount) b = 0.5 (half life decay factor) ** Half life is every 6 hours t = 75 hours Amount left = 100(0.5)75/6

  32. Continuously Compounded Interest Using the compound interest formula, evaluate (1 + 1/n)n for n = 2, 8, 32, 128, and 512. What do you think would happen as n continues to increase? Now find the value of e. (ex)

  33. Continuous Interest As the compound interest formula approaches ∞, it nears the value of e. Thus, the continuously compounded interest formula is A = Pert

  34. Compound Interest Example • Suppose you invest $1050 at an annual interest rate of 5.5% compounded continuously. Find the amount you will have in the account after 5 years. • How long will you need to leave the money in the account to double your initial investment? • Suppose you invest $1300 at an annual interest rate of 4.3% compounded continuously. How much will you have after 3 years?

  35. More Examples/ Applications Summary Questions

  36. Assignment Section 7.2 • 316 #16 – 22 Even 31 - 33,35 - 39

  37. Warm-up 7.3 Identify each function as exponential growth or decay & identify % change. • Y = 15(1.45) x 2) y = 7.3(0.8)x • Describe a function that would be modeled by exponential growth. • A radioactive element has a half-life of 30 hours. Write the exponential decay function for the 100-mg sample. Use the function to find the amount of the element remaining after 100 hours.

  38. Did you know? • Sharks are immune to cancer • Manicuring the nails has been done by people for more than 4,000 years • The study of the iris of the eye is called iridology • Back in 1919, the Russian transplant pioneer Serge Voronoff made headlines by grafting monkey testicles onto human males’. • In 1946, the New York Yankees became the first baseball team to travel by plane • By recycling just one glass bottle, the amount of energy that is being saved is enough to light a 100 watt bulb for four hours

  39. The Mall of America, located in Bloomington, Minnesota is so big that it can hold 24,336 school buses • If you have three quarters, four dimes, and four pennies, you have$1.19. You also have the largest amount of money in coins without being able to make change for a dollar. • In the United States, poisoning is the fourth leading cause of death among children • In 1916, Charlie Chaplin was making $10,000 a week, making him the highest paid actor of his time • It's possible to lead a cow upstairs...but not downstairs. • People that smoke have 10 times as many wrinkles as a person that does not smoke • Thomas Edison was afraid of the dark. (Hence, the light bulb?)

  40. Assignment 10 minutes; compare with a partner; Stay with partner, you receive “hw grade” if you get question correct when called. Section 7.2 • 316 #16 – 22 Even 31 - 33,35 - 39

  41. Summary Questions More Examples/ Applications

  42. Assignment • Buying your first new car • Financial Futures • Half-Life Handout

  43. Homework 7.1-7.2 Quiz

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