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. (7/4)(x+2) ) Write the quadratic function in the vertex form a) y = 2x x + 5 b) y = -3x x - 8 2(x+5) 2-45; (b) -3(x-3) Solve the problem. 3) The price p dollars and the quantity x sold of a certain product obey the demand equation p = -1 x + 30, 0 x (a) Express the revenue R as a function of x. Give the restricted domain. Sketch graph and label n context. (b) What quantity x maximizes the revenue? (c) What is the maximum revenue? (d) What price should the company charge to maximize revenue? (e) Express the revenue R as a function of p. Give the restricted domail. Sketch the graph and label in context. (a) R(x) = -x x dollars, 0 x 450; (b) x = 225 units; (c) R = $3375; (d) p = $15 4) 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. 2 in. 8 sq.in. 5) A developer wants to enclose a rectangular grassy lot that borders a river with 320 feet of fencing. There will be 4 fences perpendicular to the river and two parallel to the river. Write the area as a function of x, the side perpendicular to the river. What is the restricted domain? What is the largest area that can be enclosed? What are the dimensions of this optimal rectangular lot? A = 160x - 2x 2 ; (0, 80), 3200 ft 2 ; 40 ft x 80 ft. 1

2 6) 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? A = x ; x = 3.8 ft; h = 1.9 ft, smallest A = 42.9 sq. ft. x 7) An open box is constructed with a piece of cardboard 50 in. by 40 in. by cutting off squares of size x by x in each of the corners. What is the volume function? Give the restricted domail. What are the dimensions of the box of largest volume? What is the largest volume? Sketch and label in context. V = x(50-2x) (40-2x); (0, 20); L = 35.2 in, W = 25.2 in, H = 7.4 in, V = cu-in. 8) 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? C) {x 3 sec < x < 9 sec} 9) 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) What is the highest concentration? (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? (a) y = 0; (b) t = 7 hours; (c) 2.14 mg/dl; (c) after 22.8 hous 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. 10) f x = x x + 3 x - 1 Find the domain, all asymptotes, holes, if any, intercepts and graph. 11) h(x) = x + 8 x 2-1 {x x -1, 1, -8} {x x -1, 1} C) {x x 0, 1} all real numbers List the x- and y-intercepts, horizontal and vertical asymptotes and graph. 12) f(x) = -3 x - 6 2

3 Give the appropriate zeros and multiplicities. Write the function if the y-intercept is Use graph to solve f(x) > 0. 13) C) -1 (multiplicity 2), 5 (multiplicity 3); y = (6/5)(x+1) 2 (x-5) 3 ; x < -1 or x > 5 Write two cubic functions with the given zeros. 14) -5, 2, -6 f x = a(x+5)(x-2)(x+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. 15) f(x) = (x )4 (x - 6) 3-1, multiplicity 4, touches x-axis; 6, multiplicity 3, crosses x-axis with an inflection point. 2 C) State the domain, vertical and horizontal asymptote of the rational function, x- and y- intercepts. 16) f(x) = x - 1 x2 + 5 _ (-, ); no vertical asymptote, y=0 is the HA; (0, -1/5), (1, 0) C) 3

4 Write a cubic function with the given zeros. 17) Write the cubic function if the zeros are: -2, 6, -6and the y-intercept is 8. f x = (-1/9) (x 3 + 2x 2-36x - 72) Analyze the graph of the given function f as follows: (a) Determine the end behavior: find the power function that the graph of f resembles for large values of x. (b) Find the x- and y-intercepts of the graph. (c) Determine whether the graph crosses or touches the x-axis at each x-intercept. (d) Graph f using a graphing utility. (e) Use the graph to determine the local maxima and local minima, if any exist. Round turning points to two decimal places. (f) Use the information obtained in (a) - (e) to draw a complete graph of f by hand. Label all intercepts and turning points. (g) Find the domain of f. Use the graph to find the range of f. (h) Use the graph to determine where f is increasing and where f is decreasing. 18) f(x) = -x 2 (x - 1)(x + 3) (a) For large values of x, the graph of f(x) will resemble the graph of y = -x 4. (b) y-intercept: (0, 0), x-intercepts: (-3, 0), (0, 0), and (1, 0) (c) The graph of f crosses the x-axis at (1, 0) and (-3, 0) and touches the x-axis at (0, 0). (e) Local maxima at (-2.19, 12.39) and (0.69, 0.55); Local minimum at (0, 0) (f) (g) Domain of f: all real numbers; range of f: (-, 12.39] (h) f is increasing on (-, -2.19) and (0, 0.69); f is decreasing on (-2.19, 0) and (0.69, ) 4

5 Make up a rational function that has the given graph. Note that the y-intercept is 1/3. 19) Also, describe end and local behaviors using symbols. R(x) = For the given function, find all asymptotes: vertical, horizontal. 20) f(x) = 6x x2-9, x - 2 (x + 2)(x - 3) y = 6; x = 3, x = -3 21) f(x) = x - 1 x2 + 2 No vertical, y = 0 22) f(x) = x 2 + 2x + 4 x + 9 y = x - 7; x = -9 None 23) f(x) = x - 6 x2-4, x = 2, x = -2; y = 0 x = -2 C) x = 2 x = 6 5

6 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 {x x 6, x 8} This function does not have any holes because the numerator and denominator do not have a common linear factor. (0, -5/12); x = -5, and 4 C) 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 (-, 1] or [3, ) Explain your choice. 26) Identify the leading term of the following polynomial function. Describe end behavior using symbols. _ f(x) = -2x 6 f(x) = -2x 5 C) f(x) = -2x 4 f(x) = -2x 7 Use the given zero to find the remaining zeros of the function. 27) f(x) = x4-45x2-196; zero: -2i 2i, 7, -7 2i, 14i, -14i C) 2i, 14, -14 2i, 7i, -7i Analyze the graph of the rational function. 28) 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. C) vertical asymptote: x = -2; hole at 3, 7 5 ; HA: y = 1; x-intercept is x = -4 6

7 Solve the inequality. x ) (x + 9) 2 < 0 (-, -9) U (-9, 2) _ For the polynomial, one zero is given. Find all others. 30) P(x) = x4-5x2-36; -2i _ 2i, 3, -3 Analyze the graph of the rational function. 31) 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 C) vertical asymptote: x = 6; hole at -5, 9 11 ; HA: y = 1; x-intercept at x = 4; y-intercept = 2/3 For the polynomial, find all the zeros and classify them as real or imaginary. Which ones are the x-intercepts? 32) P(x) = (x 2 + 6) (x 2-16) (x + 3) 7

8 Answer the question. 33) 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. (a) The x-intercepts are 2, 3, and 4. The y-intercept is 288. (b) The graph crosses the x-axis at 2 and 4, (at x=2 there is an inflection point) and touches it at 3. (c) The graph resembles f(x) = x 6 for large values of x. 34) Solve the inequality 2x + 5 x - 4 3x C), -5/2 5/3, 4 Construct a rational expression with the given characteristics. 35) 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 = (x + 1)(x + 4)2 R(x) = (x + 2) 2, (x - 3) Determine the intervals where the function is positive. 36) f(x) = (x+5) 2 (x - 4) (x - 8) (-, -5)(-5, 4) (8, ) _ 8

9 Form a polynomial whose zeros and degrees are given. 37) Zeros: -3, multiplicity 2; 1, multiplicity 1; 5, multiplicity 3; degree = 6 P(x) = (x + 3) 2 (x - 1)(x - 5) 3 Construct a polynomial with the given properties. 38) 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. C) P(x) = (-1/3)(x + 2)(x - 3)(x - 5) 2 Determine the x values that cause the function to be (a) zero, (b) undefined, (c) positive, and (d) negative. 39) f x = x - 6 2x + 1 C) (a) 6, (b) (-1/2), (c), -1/2 6,, (d) -1/2, 6 40) 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 y = ((x-3)(3x-2))/(5(x-3)(x+2)) Find the best model that fits the data. 41) 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. y = 0.03x x x

10 Solve the problem. 42) A can in the shape of a right circular cylinder is required to have a volume of 700 cubic centimeters. The top and bottom are made up of a material that costs 8 per square centimeter, while the sides are made of material that costs 5 per square centimeter. Which function below describes the total cost of the material as a function of the radius r of the cylinder? C(r) = 0.08 r C(r) = 0.16 r r r C) C(r) = 0.16 r r C(r) = 0.08 r r 43) A closed box with a square base has to have a volume of 18,000 cubic inches. Find a function for the surface area of the box. S(x) = 2x ,000 x C) S(x) = x ,000 x 44) Which of the following functions could have this graph? S(x) = 2x ,000 x S(x) = 2x ,000 x y = 2(x - 2)2 (x - 6) (x + 1)(x - 4) 2 y = (x - 2)2 (x - 6) (x + 1)(x - 4)2 (x + 1)(x - 4) 2 C) y = (x - 2)2(x - 6) y = (x - 2)(x - 6)2 (x + 1)2(x - 4) 10