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Advanced Algebra with Trigonometry Unit 2 Take-Home Test Name Directions: Show all work and reasoning to receive full credit. For Questions 1 & 2, determine the domain of the functions. Answers should be in interval notation. x4 1) h( x) 3 2) g ( x) 3 7 x x 4x 3) Determine and simplify the difference quotient f, f ( x h) f ( x ) , h 0 , given f ( x) 2 x2 9 x 5 . h 4) Answer the following questions about the function f (x) 3x 6 x2 4 5 a) If f (x) , what is x? What point(s) are on the graph? 3 b) List the x-intercepts and y-intercept, if any, of the graph of f. Check your answers. For Questions 5 & 6, determine algebraically whether the function is even, odd, or neither. Then state any symmetry. x3 3 5) f ( x) 2 6) g ( x) 3 x 2 8 4x 9 7) Use a graphing utility to graph the function f ( x) 2 x 3 7 x 2 3x . a) Determine to the nearest thousandths any relative minimum and maximum values. b) Determine the open interval(s) on which the function is increasing, decreasing, or constant. 8) Find the average rate of change of the function f (x) x3 from -3 to 2. (No decimal approximations). x2 3 9) A marina owner wishes to estimate a linear function that relates boat length in feet and its draft (depth of boat below water line) in feet. He collects the following data. Let boat length represent the independent variable and draft represent the dependent variable. Use a graphing utility to draw a scatter plot and to find the line of best fit. Use the equation to determine the draft for a boat 60 ft in length (to the nearest thousandth). Boat Length (ft) 25 25 30 30 45 45 50 50 Draft (ft) 2.5 2 3 3.5 6 7 7 8 10) Write a rule for piecewise function graphed. f ( x) 1 1 12) Graph by hand the following function. List, in order, the sequence of transformations. 1 f ( x) x 2 3 2 1 11) Graph the given piecewise function. 1 2 x 2, (, 2] 1 f ( x) x 2 3, (2, 4] 2 3x 13, (4, ) 1 1 13) Graph by hand the following function. List, in order, the sequence of transformations. g ( x) 2 x 3 1 1 1 14) Determine the equation of the following graph. 1 1 15) Determine the equation of the following graph. 1 1 1 16) An open box is made from a rectangular piece of material by cutting equal squares from each corner and turning up the sides. Write the volume of the box as a function of x if the material is 24 inches by 16 inches. Simplify and write your equation in standard form. Use a graphing utility to determine the value of x that will maximize the volume of the box.