Warm-up 2/26/09

1. Warm-up 2/26/09 SKILL REVIEW: Simplify each expression. Use only positive exponents. • (3a2)(4a6) • 8a5/2a2 • (-6m2n2)(3mn) • X4X-2/x-5 • (p2q)-2 • (s2t)3(st)

2. Good News Chapter 5 projects changed to be due on Tuesday 

3. Did you know? • The reason why flamingos are pink is because they eat shrimp which have a red pigment • One million cloud droplets are needed to make enough water to produce one raindrop. • Rice flour was used to strengthen some of the bricks that make up the Great Wall of China • Research has indicated that a tie that is on too tight can increase the risk of glaucoma in men • Each year all of the Hostess bakeries combined bake 500 million Twinkies a year. (A twinkie is a sponge cake with a creamy filling.)

4. Pluto is the only planet in our solar system that has not been visited by a spacecraft • The worlds tallest free fall rollercoaster is The Giant Drop located in Australia. The drops is 120 meters which is equivalent to a 39 storey building • Stalks of sugar cane can reach up to 30 feet • The markings that are found on dice are called "pips." • A newly hatched fish is called a "fry." • The music band UB40 got its name from an unemployment form in England • Armadillos can be housebroken • The Eisenhower interstate system requires that one-mile in every five must be straight. These straight sections are usable as airstrips in times of war or other emergencies • The material to build the TajMahal was brought in from various parts of India by a fleet of 1000 elephants

5. Chapter 5 Project • Unit map

12. §6.1 : Power Functions and Inverses LEQ: What are the similarities and differences between odd and even functions? LEQ: What are the general shapes of functions? Sketch a graph of y = (x + 1)2 + 2 and its inverse. Determine if each graph depicts a function.

13. Functions like y = x4 and w = .084C3 are power functions. Power function – A function with the form y = axn, where a≠0 and n is a positive integer. Even functions – Have the y-axis as the axis of symmetry Odd functions – have the origin as the point of symmetry. *A graph has point symmetry if there is a ½ turn (rotation of 180 deg) that maps the graph onto itself.

14. Try these Graph each function. Describe its symmetry. Tell whether it is even, odd, or neither. • Y = x5 • Y = IxI • Y = 2x + 3 • Y = 2x4

15. Inverses To undo a square, you square root. To unto a cube, you __________. To undo an exponent of 4, you ________. You can use a radical symbol or a rational exponent to indicate roots.

16. Ex. y = 5√32 index (the root you want) y = 321/5 The principal root is the positive root. Ex. 4√81 = +9, -9 +9 principal root Rational Exponent Property bm/n = n√bm

17. Simplify each expression • 82/3 • 43/2 • (53/4)4/3 • 2434/5 • 4 • 8 • 5 • 81

18. Inverses Find the inverse of y = 2x4 x = 2y4 x/2 = y4 (x/2)1/4 = y4(1/4) f’(x) = (x/2)1/4 Is the inverse of y = 2x4 a function? (graph) The inverse is not a function – it does not pass the vertical line test.

19. Extraneous Solution A solution is extraneous if: -It satisfies the end equation -It does NOT satisfy the initial equation *To eliminate extraneous solutions, you should check each possible solution in the original equation.

20. Ex Solve √x = x – 6 and discard extraneous solutions. (√x)2 = (x – 6)2 x = (x – 6)2 x = x2 – 12x + 36 0 = x2 – 13x + 36 0 = (x – 4)(x – 9) x = 4 or x = 9

21. Check solutions Solution x = 4 √x = x – 6 √4 = 4 – 6 2 ≠ -2 Not a solution 4 is extraneous Solution x = 9 √x = x – 6 √9 = 9 – 6 3 = 3 9 is the only solution

22. T.O.T.D. What are the similarities and differences between odd and even functions? Create an example problem that would have an extraneous answer.

23. Assignment/HW Section 5.1 p.259 #2-50 Even

24. Warm-up 2/27/09 Simplify each function • 243(1/5) • 81(3/4) • Find the inverse of y = x2 Graph the function and its inverse on the same graph.

25. Did you know? • Medical research has found substances in mistletoe that can slow down tumor growth • In the USA, 32% of employees eat lunch and work at the same time • In Alabama, it is against the law to wear a fake mustache that could cause laughter in the church • In the United States, more than 4.2 million couples live together that are not married • Bill Gates house was partially designed using a Macintosh computerThe most expensive perfume in the world is Parfum VI, which was made by Arthur Burnham. A 4 inch bottle which is covered with diamonds and 24-carat gold costs $71,380 • If Wal-Mart was classified as a country, it would be the 24th most productive country in the world • In Australia, a dust-devil is called a "willy-willy

26. The male howler monkey of Central and South America is the noisiest land animal, which can be heard clearly from a distance of ten miles away • Nerve cells can travel as fast as 120 metes per second • It is said that grapefruit got its name because it grows like grapes in clusters. One cluster can have up to 25 grapefruits • An average adult produces about half a liter of flatulent gas per day, resulting in an average of about fourteen occurrences of flatulence a day • Peanut butter is an effective way to remove chewing gum from hair or clothes • The longest kiss on record lasted 30 hours and 45 minutes. Dror Orpaz and Carmit Tsubara recorded it on April 5, 1999 at a kissing contest held in Tel Aviv, Israel • Polar bears are excellent swimmers. They have been known to swim more than 60 miles without a rest • The Leaning Tower of Pisa is 58.36 meters above the ground

27. 15 minutes to revise, edit, discuss Assignment/HW Section 5.1 p.259 #2-50 Even Swap papers, Check answers Circle # missed at the top. How did you do? Less than 5 missed? You’ve got this. More than 8 missed? You need to work on this.

28. §6.2: Polynomial Functions LEQ: How can you describe the graphs of polynomial functions? LEQ: What are the end behaviors of polynomial functions? Work Together p.261 Groups of 2-3

29. Polynomial Function – Any sum or difference of power functions and constants. Degree – The exponent determines the degree of the function. Standard form – All like terms are combined. The function is written in descending order by degree.

30. Describing Polynomials

31. Classify each polynomial Name & degree: • 5x2 – 7x • -10x3 • X2 + 3x - 2 • binomial, 2 • Monomial, 3 • Trinomial, 2

32. End Behavior The end behavior of a graph describes the far left and the far right part of the graph. Four types of behavior: Up, Up Up, Down Down, UP Down, Down Ex. Find a graph that represents each type of behavior.

33. Between two endpoints of a graph: - a polynomial of degree “n” will have exactly one y-intercept and up to “n” x-intercepts -Use a table or graph to describe end behavior and estimate intercepts. Describe the end behavior and estimate the x- and y-intercepts. Ex. Y = x3 – 3x2 + x - 2

34. How do you know without a calculator? • Everything can be simplified to the two most basic graphs: linear and quadratic • Even graphs: Both ends will go up, or both will go down? • How do you know which it will be? • Both up: Highest term (even) is Positive • Both down: Highest term is Negative • Odd powered graphs: One end goes up and one end goes down • Think about linear graphs (positive, negative) as your basis.

35. Practice:Describe the end behavior: (without graphing) • x3 + 6x2 – x – 6 • X4 – 5x3 • X3 – 8x • X4 – 4x3 – 4x2 + 16x • -2t3 - 3t2 + 8t + 12

36. Summarize What You Know • What are some ways you can describe polynomial functions? • How can you determine the “end behavior” of a graph? • How do you write a function in standard form? • How do you determine the degree? • How can you quickly determine the maximum number of “answers”/x-intercepts an equation will have?

37. Assignment On a piece of paper, to turn in: p.264 #13, p. 267 Checkpoint #1 - 9

38. Homework • P. 263 - 264 • 1 – 37 odd • Answers are in back, • check yours! • (Don’t just “copy”!)

39. Warm-up 10/8/08 Find the inverse of each function. Graph the function and its inverse. • Y = 5x4 2) y = 23√x 3) y = (x + 1)3 Write the polynomial in standard form and classify by degree and # of terms: • N = 6m4 – m + 4m2 5) f(t) = 2t + 5t3 – 9 • What is the end behavior of f(x) = 4x6+ x3? • When does a graph have point symmetry?

40. Food for Thought: Did you know? http://www.gapsc.com/msgc/msgc_compelling.asp http://www.lps.k12.co.us/schools/arapahoe/fisch/fischbowlpresentations.htm http://thefischbowl.blogspot.com/2007/06/did-you-know-20.html Have you seen it?

41. Interesting Facts • A species of dolphin is born naturally blind in the Indus and Ganges rivers in South Asia. These dolphins have a highly sophisticated sonar system and swim on only one side of their body • Kermit the Frog was named after Kermit Scott, a childhood friend of creator Jim Henson, who became a professor of philosophy at Purdue University • Weatherman Willard Scott was the first Ronald McDonald

42. Aztec emperor Montezuma had a nephew, Cuitlahac, whose name meant "plenty of excrement." • Hang On Sloopy is the official rock song of Ohio. • Actor Sylvester Stallone once had a job as a lion cage cleaner • Play-Doh was introduced in 1956 by Hasbro Inc. The only color availabe was an off white, and it came in one size which was a one and a half pound can • The USSR launched the world's first artificial satellite, Sputnik 1, in 1957. • An oyster can change its gender

43. 6.1 – 6.2 Quiz • You can work with A partner (no more) or alone. • Graphs must be accurate – locate vertex, etc. • 20 minutes…

45. §6.3: Polynomials & Linear Functions Sometimes its more useful to work with polynomials in factored form: Ex. X3 + 6x2 + 11x + 6 = (x + 1)(x + 2)(x + 3) How could you prove these are factors of the polynomial?

46. Maximum & Minimum? In many polynomial functions, there is no absolute maximum or minimum. However, there may be a relative maximum or relative minimum. (compared to the points around them, they are the maximum or minimum)

47. If a function is in factored form, you can use the zero product property to find the values that will make the function equal zero. Ex. F(x) = (x – 1)(x + 2)(x – 4) For what values of x will f(x) = 0? 1, -2, 4

48. Factor Theorem The expression (x – a) is a linear factor of a polynomial if and only if the value “a” is a zero of the related polynomial function. Ex. Graph y = x2 + 3x – 4 It crosses the x-axis at 1. Thus, (x – 1) is a factor of the polynomial. Also (x + 4) is a factor of the polynomial.

49. What the answer tells you: In the previous example, (x – 1) and (x + 4) were the factors of the polynomial. Therefore: 1) -4 is a solution of x2 + 3x – 4 2) -4 is an x-intercept of y = x2 + 3x – 4 3) -4 is a zero of y = x2 + 3x – 4 4) x + 4 is a factor of y = x2 + 3x - 4

50. Ex) Determine the zeros of the function y = (x – 2)(x + 1)(x + 3) x = 2, -1, -3 Ex) Write four equivalent statements about one of the solutions of the equation x2 – 4x + 3 = 0 Factors (x – 3)(x – 4)