B Apply

40. Think About a Plan The data shows the power generated by a wind turbine. The x column gives the wind speed in meters per second. The y column gives the power generated in kilowatts. What is the degree of the polynomial function that models the data?

    x	y
    5	10
    6	17.28
    7	27.44
    8	40.96
    9	58.32

    • What are the first differences of the y-values?
    • What are the second differences of the y-values?
    • When are the differences constant?

Classify each polynomial by degree and by number of terms. Simplify first if necessary.

41. a 2 + a 3 − 4 a 4 eh squared , plus , eh cubed , minus , 4 eh to the fourth
42. 7
43. 2 x ( 3 x ) 2 x open 3 x close
44. ( 2 a − 5 ) ( a 2 − 1 ) open 2 eh minus 5 close open , eh squared , minus 1 close
45. ( − 8 d 3 − 7 ) + ( − d 3 − 6 ) open negative 8 , d cubed , minus 7 close plus open negative , d cubed , minus 6 close
46. b ( b − 3 ) 2 b open b minus 3 , close squared

Determine the sign of the leading coefficient and the least possible degree of the polynomial function for each graph.

47. 
48. 
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50. Open-Ended Write an equation for a polynomial function that has three turning points and end behavior up and up.
51. Show that the third differences of a polynomial function of degree 3 are nonzero and constant. First, use f ( x ) = x 3 − 3 x 2 − 2 x − 6 . f open x close equals , x cubed , minus , 3 x squared , minus 2 x minus 6 .  Then show third differences are nonzero and constant for f ( x ) = a x 3 + b x 2 + c x + d , a ≠ 0 . f open x close equals eh , x cubed , plus b , x squared , plus c x plus d comma eh not equal to 0 .

52. Reasoning Suppose that a function pairs elements from set A with elements from set B. A function is called onto if it pairs every element in B with at least one element in A. For each type of polynomial function, and for each set B, determine whether the function is always, sometimes, or never onto.

    1. linear; B = b equals  all real numbers
    2. quadratic; B = b equals  all real numbers
    3. quadratic; B = b equals  all real numbers greater than or equal to 4
    4. cubic; B = b equals  all real numbers

53. Make a table of second differences for each polynomial function. Using your tables, make a conjecture about the second differences of quadratic functions.

    1. y = 2 x 2 y equals , 2 x squared
    2. y = 5 x 2 y equals , 5 x squared
    3. y = 5 x 2 − 2 y equals , 5 x squared , minus 2
    4. y = 7 x 2 y equals , 7 x squared
    5. y = 7 x 2 + 1 y equals , 7 x squared , plus 1
    6. y = 7 x 2 + 3 x + 1 y equals , 7 x squared , plus 3 x plus 1

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