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Unit 6 Summary Sheet Name: Determine whether or not each of the following is a polynomial function. If not, state why. If so, state the type of polynomial, the degree, and the leading coefficient. Then, write the function in standard form. 4 1) f ( x) 3x 5x 2 x 2 x 2 3 2) g ( x) 3x 5x 3x 2 4 3) h( x) 8x 6x 2 5x3 4) Evaluate f (3) for the function f ( x) 2 x3 2 x 2 x 5) Factor the function f ( x) 2 x3 3x 2 17 x 30 given that x 3 is one factor. 6) State the end behavior for the functions: a) g ( x) 3x 4 10 x3 8 x 2 b) j (n) 3n5 4n3 2n2 12 7) List all possible rational roots for the function h( x) 6 x3 4 x 2 2 x 10 . 8) Find all zeros for the function k ( x) 2 x 4 3x3 26 x 2 21x 90 2 3 9) Find all zeros for the function h( x) x3 x 2 3x 10 Graph each of the following. Label all intercepts. 10) f ( x) x 2 x 3 x 4 x 2 11) g ( x) x 1 x 2 2 Write the equation for each of the functions graphed below. 12)________________________________ 13)______________________________ 14) A garden measuring 12 meters by 16 meters is to have a pedestrian pathway installed all around it, increasing the total area to 285 square meters. What will be the width of the pathway? 15) You are given a rectangular sheet of cardboard with dimensions 20 inches by 30 inches. You need to make an open-topped box out of the cardboard by cutting squares out of the corners and folding up the sides. a) Find the length of the sides of the squares you must cut out to obtain the maximum volume possible for the box. b) What is the maximum volume possible for the box? Window Hint: Xmin: 0 Xmax: 10 Ymin: 0 Ymax: 1200 16) Consider the function f ( x) 2 x 4 x 5 x 1 a) What are the relative extremum for the function? b) State the intervals where the function is increasing. c) State the intervals where the function is decreasing. 17) Select all of the following that have an end behavior of: as x , f ( x) . a) f ( x) x3 4 x 2 5x 6 b) f ( x) x 4 2 x c) f ( x) x5 x3 4 x 1 d) f ( x) 2( x 3) 2 5 e) f ( x) 3( x 5)3 1 f) f ( x) 1 x 5 2 18) The data in the chart represents the height of a football that is kicked off the ground. Time 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 (seconds) Height 0 10 16 22 24 25 23 20 16 (feet) a) Use your calculator to graph the data and find the equation of the curve of best fit for the data. Round your answer to 3 decimal places. b) Explain why you chose the model that you chose in (a). c) Use your equation to find when the football will hit the ground. Explain how you determined your answer. 19) Write the function of least degree with a leading coefficient of 1 that has the zeros x = -3, x = 4, x = 2i, and x = -2i. Write your answer in standard form. 20) What does the Fundamental Theorem of Algebra tell us about a polynomial function? 21) How do you determine the end behavior of an nth degree polynomial function? 22) How can real-life problem situations be modeled by polynomial functions? 23) When given a graph of a polynomial function, how can you tell if there are any double zeros? 2012-2013