Remember to SHOW ALL STEPS. You must be able to solve analytically. Answers are shown after each problem under A, B, C, or D.

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1 Math Review Chapters 3 and 4 Name Remember to SHOW ALL STEPS. You must be able to solve analytically. Answers are shown after each problem under A, B, C, or D. Find the quadratic function satisfying the given conditions 1) Vertex V(-2, 3) and y-intercept of 10. 2) Write the quadratic function in the vertex form a) y = 2x x + 5 b) y = -3x x - 8 3) Write a quadratic function if one of the zeros is x = 1 - i Solve the problem. 4) The price p dollars and the quantity x sold of a certain product obey the demand equation x = -15p + 450, 0 p 30. (a) Express the revenue R as a function of x. Give the restricted domain. Sketch graph and label. (b) What quantity x maximizes the revenue? (c) What is the maximum revenue? (d) What price should the company charge to maximize revenue? 5) A piece of rectangular sheet metal is 8 inches wide. It is to be made into a rain gutter by turning up equal edges to form parallel sides. Let x represent the length of each of the parallel sides. For what value of x will the area of the cross section be a maximum (and thus maximize the amount of water that the gutter will hold)? What is the maximum area? Sketch the area function and label. 6) A developer wants to enclose a rectangular grassy lot with 320 feet of fencing. Write the area function with the restricted domain. What is the largest area that can be enclosed? What are the dimensions of this optimal rectangular lot? 7) An open box with a square base is required to have a volume of 27 cubic feet. Express the amount A of material used to make such a box as a function of the length x of a side of the base. What are the dimensions of the box with smallest surface area. What is the smallest surface area? 8) An open box with a square base is constructed with a piece of cardboard 50 in. by 40 in. What is the volume function? What are the dimensions of the box of largest area/ What is the largest volume? sketch and label. 9) A ball is thrown vertically upward with an initial velocity of 192 feet per second. The distance in feet of the ball from the ground after t seconds is s = 192t - 16t 2. For what interval of time is the ball more than 432 above the ground? 1

2 10) The concentration C of a certain drug, (in mg/dl) in a patient's bloodstream is given by C(t) = 30t t 2., t hours after the drug was given + 49 (a) Find the horizontal asymptote of C(t). What does it represent in the situation? (b) Using a graphing utility, determine the time at which the concentration is highest. (c) In order for the drug to be effective, the concentrations should be at least 1.2 mg/dl.; when should the medicine be taken again? Give the appropriate zeros and multiplicities. Write the function if the y-intercept is Use graph to solve f(x) > 0. 11) Graph the function without the calculator. What is the degree? What is the end behavior?, describe with arrows and with symbols What are the zeros? Specify the multiplicity of each zero and indicate the behavior of the graph at each x-intercept: crosses, bounces or crosses with an inflection point. 12) f(x) = -2x(x-1)(x - 2) 13) f x = x x + 3 x - 1 List the x- and y-intercepts, horizontal and vertical asymptotes and graph. 14) f(x) = -3 x - 6 2

3 Make up a rational function that has the given graph. Note that the y-intercept is 1/3. 15) Describe end and local behaviors using symbols. Write two cubic functions with the given zeros. 16) -5, 2, -6 For the polynomial, list each real zero and its multiplicity. Determine the behavior of the graph at each x-intercept: crosses, bounces or crosses with an inflection point. Sketch the graph. 17) f(x) = (x )4 (x - 6) 3 Write a cubic function with the given zeros. 18) Write the cubic function if the zeros are: -2, 6, -6and the y-intercept is 8. State the domain, vertical and horizontal asymptote of the rational function, x- and y- intercepts. 19) f(x) = x - 1 x2 + 5 For the given function, find all asymptotes: vertical, horizontal or oblique 20) f(x) = 9x x2-6, 21) f(x) = x - 1 x ) f(x) = x 2 + 2x + 4 x ) f(x) = x - 6 x2-4, 3

4 Give the domain of the function. Does the function contain any holes? Explain. What are the x- and y-intercepts? 24) R(x) = x2 + x - 20 x x + 48 Solve the inequality using the "signs" method, then graph the function without the calculator to check your answer. Use interval notation. (x - 1)(3 - x) 25) (x - 2) 2 0 Match the graph with the polynomial function. Explain your choice. 26) Describe end behavior using symbols. A) f(x) = -x6 + 7x5 - x2-2x + 16 B) f(x) = -2x5 + 7x4 + 9x3-40x2 + 4x + 16 C) f(x) = 2x6 + 9x3-7x2 + 4x - 16 D) f(x) = x5 + 7x4 - x3-40x2 + 2x + 16 Analyze the graph of the rational function. 27) Find the vertical, horizontal asymptote(s), coordinates of any hole, if any, intercepts and graph without the calculator. R(x) = x2 + x - 12 x 2 - x - 6. Describe end and local behavior using symbols. Solve the inequality. x ) (x + 9) 2 < 0 For the polynomial, one zero is given. Find all others. 29) P(x) = x4-5x2-36; -2i Analyze the graph of the rational function. 30) Find the vertical, horizontal asymptote(s), coordinates of any hole, if any, intercepts and graph without the calculator. R(x) = x2 + x - 20 x 2. Describe end and local behavior using symbols. - x - 30 For the polynomial, one zero is given. Find all others. 31) P(x) = x3 + 3x2-8x + 10; 1 + i 4

5 Answer the question. 32) For the polynomial f(x) = (x - 2) 3 (x - 3) 2 (x - 4) (a) Find the x- and y-intercepts of the graph of f. (b) Determine whether the graph crosses, touches or has an inflection point at each x-intercept. (c) End behavior: find the power function that the graph of f resembles for large values of x. (d) Graph without the calculator. 33) Which of the following polynomial functions might have the graph shown in the illustration below? A) f(x) = x 2 (x - 2) 2 (x - 1) 2 B) f(x) = x(x - 2) 2 (x - 1) C) f(x) = x 2 (x - 2)(x - 1) D) f(x) = x(x - 2)(x - 1) 2 Find the real solutions of the equation. 34) 2x 3 - x 2-20x + 10 = 0 35) Solve the inequality 2x + 5 x - 4 3x Construct a rational expression with the given characteristics. 36) The graph of R(x) crosses the x-axis at -1, touches the x-axis at -4, has vertical asymptotes at x = -2 and x = 3, and has one horizontal asymptote at y = -2. Determine the intervals where the function is positive. 37) f(x) = (x+5) 2 (x - 4) (x - 8) Form a polynomial whose zeros and degrees are given. 38) Zeros: -3, multiplicity 2; 1, multiplicity 1; 5, multiplicity 3; degree = 6 Find all the zeros of the polynomial P. 39) P(x) = 2x 4-2x 3 + x 2-5x - 10 Form a polynomial f(x) with real coefficients having the given degree and zeros. 40) Find a third-degree polynomial function with real coefficients and with zeros 1 and 3 + i. 5

6 Information is given about a complex polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. 41) Degree 6; zeros: -2, 2 + i, -3 - i, 0 Construct a polynomial with the given properties. 42) The graph of the polynomial crosses the x-axis at -2 and 3, touches the x-axis at 5, crosses the y-axis at 50 and is below the x-axis between -2 and 3. Determine the x values that cause the function to be (a) zero, (b) undefined, (c) positive, and (d) negative. 43) f x = x - 6 2x ) Write a rational function with ALLthe following characteristics a) The domain is all real numbers except -2 and 3 b) It has an x-intercept at 2/3 c) It has a hole at x = 3 d) The horizontal asymptote is y = 3/5 Write the sum or difference in the standard form a + bi. 45) (2-4i) + (4 + 2i) 46) (5 + 6i) - (-7 + i) Write the product in standard form. 47) (8 + 9i)(8-9i) Find the best model that fits the data. 48) The profits (in million dollars) for a company for 8 years was as follows: Year, x Profits 1993, , , , , , , , Find the cubic function of best fit to the data. 6