MATH 101 EXAM 3 REVIEW I Quadratic Functions Answer the following questions for each of the quadratic functions given 1 f x x2 6x 1 a b c d e f g h 2 f x 2x2 7x 9 3 f x x2 3x 6 Find the vertex of the graph of f x Be able to show work What is the equation of the axis of symmetry Identify the y intercept of the graph of f x Does the function have a maximum or a minimum Make a careful labeled sketch of the graph of f x Use the symmetry property Over what interval s is the graph increasing decreasing What is the range of the graph of f x Find the exact zeros of the function analytically Verify by using a graphing calculator 4 Show how to use completing the square to put 1 above in the form f x a x h 2 k 5 Determine the quadratic function with vertex at 1 4 and passing through 5 5 Write your answer in the form f x ax2 bx c 6 The profit that a vendor makes per day by selling x pretzels is given by the function P x 0 002x2 1 2x 150 Find the number of pretzels that must be sold to maximize profit What is the maximum profit 7 A developer wants to convert a rectangular grassy lot that borders a city street to a parking area If the developer has 284 feet of fencing to enclose the parking area and does not fence the side along the street what is the largest area that can be enclosed What are the dimensions of the lot 8 The quadratic function f x 0 041x2 0 45x 30 06 models the median age at which men in the United States were first married x years after 1990 In which year was this average age at a minimum Round to the nearest year What was the average age at first marriage for that year Round to the nearest tenth 9 If a football is kicked straight up with an initial velocity of 64 ft sec from a height of 5 ft then its height above the earth is a function of time given by h t 16t2 64t 5 a What is the maximum height reached by the ball b When will the ball be at its highest point c What is the hang time for the kick that is when will the ball hit the ground II Higher degree Polynomial Functions 10 f x 5 x 1 2 x 5 x 2 3 is a polynomial function a What is the degree of P x b Describe the left and right end behavior of the graph c List each real zero and its multiplicity Then determine whether the graph crosses or touches the x axis at each x intercept 11 Use a graphing calculator to find the real zeros if any and the absolute or local maximum or minimum value s for the function below Round to two decimal places if needed f x 35x4 1 48x3 1 75x 2 62 12 Show either long division or synthetic division to find the quotient when the polynomial P x 8x4 23x3 3x2 x 4 is divided by x 3 Write P x in the form quotient divisor remainder 13 Is x 1 a factor of x3 2x2 3x 2 Explain 14 Is 1 a zero of 2x4 x3 4x2 3x 1 Explain 15 List all the potential rational zeros for the polynomial f x 6x4 4x3 2x2 15 16 Use the given zero to find all the zeros of the polynomial P x x4 6x3 7x2 6x 6 with zero i 17 Find a polynomial function P x of least possible degree having real coefficients with the given zeros a 3 1 with multiplicity 2 b 2 i 5 c 1 1 3 1 3 18 Use algebraic techniques to find all the real or complex zeros of P x 3x3 5x2 3x 2 19 A rectangular piece of sheet metal measuring 30 inches by 36 inches is to be made into an open box by cutting out equal sized squares of side length x from each corner and folding up the sides a b c d e What are the restrictions on the values for x Write a function V which will give the volume of the open box Find the volume if a square that is 4 inches on a side is cut out For what value of x will the volume be a maximum What is the maximum volume for this box 20 Use a sign test to solve the following inequalities analytically Verify your solution by graphing a x3 4x2 12x 0 b x 1 x 3 x 4 0 c x 2 2 x 3 x 6 0 Answers 1 f x x2 6x 1 a V 3 10 b x 3 c 0 1 d minimum e 2 f x 2x2 7x 9 3 f x x2 3x 6 V 74 121 8 7 x 4 0 9 minimum V 32 15 4 3 x 2 0 6 maximum 20 20 20 15 15 15 10 10 10 5 5 5 20 15 10 5 5 0 5 10 15 20 15 10 5 5 10 15 f inc 3 dec 3 g 10 h x 3 10 4 f x f x 0 5 10 15 20 20 15 10 5 5 10 10 15 15 74 inc dec 74 121 8 9 x 1 2 0 5 10 15 20 inc 32 dec 32 15 4 3 i 15 x 2 x2 6x 9 9 1 x2 6x 9 9 1 x 3 2 10 5 f x 14 x 1 2 4 14 x2 12 x 15 4 6 300 pretzels 7 10 082 square feet dimensions are 71 ft by 142 ft 8 1995 28 8 years old 9 69 ft 2 seconds about 4 1 seconds 10 a Degree 6 b end behavior rise left rise right c real zeros 1 M2 touch 5 M1 cross 2 M3 cross 11 zeros 1 37 4 04 Maximum value s 8 99 absolute 3 31 local Minimum value 1 82 local 12 P x 8x3 x2 1 x 3 1 13 Yes One explanation using the Factor Theorem division by x 1 results in a remainder of 0 14 No One explanation using the Remainder Theorem substituting 1 for x does not result in a value of 0 15 possible rational zeros 1 3 5 15 21 31 16 23 52 53 56 15 2 16 Zeros i i 3 3 One possible procedure x i x i also So x i x i x2 1 is a factor of P x Divide P x by x2 1 using long division then find the zeros of the quadratic quotient 17 a P x x3 5x2 7x 3 b P x x3 9x2 25x 25 c P x x3 x2 4x 2 18 Zeros 2 1 6 13 1 13 6 19 a 0 x 15 b V …