Quiz 4 SU20 Math 107

NAME : Daniel Jun I have completed this assignment myself, working independently and not consulting anyone except the instructor. MATH 107 QUIZ 4 INSTRUCTIONS Summer 2020 Instructor: D. Zulli ● The quiz is worth 100 points. There are 13 problems. This quiz is open book and open notes . This means that you may refer to your textbook, notes, and online classroom materials, but you must work independently and may not consult anyone (and confirm this with your submission.) You may take as much time as you wish, provided you turn in your quiz no later than Wednesday, July 28 at 11:59 p.m. ● Show work/explanation where indicated. Answers without any work m ay earn little, if any, credit. You may type or write your work in your copy of the quiz, or if you prefer, create a document containing your work. Scanned work is acceptable also. In your document, be sure to include your name and the assertion of independence of work. ● If you have any questions, please contact me by e-mail diane.zulli@faculty.umuc.edu 1. (4 points) Solve the inequality 2x 2 - 14x ≤ 0 and write the solution set in interval notation. a. (-∞,0] U [7,∞) b. [0, 7] c. (-∞, 7] d. (-∞,7] U [0,∞) 2. (4 points) Solve rational expression ( x 2 −2 x x −15) > 0 and write the solution set in interval notation. a. (-∞,-3) U (0,∞) b. (5,∞) c. (-3,0) U (5,∞) d. (-∞,-3) U (0,5) 3. (4 points) For f(x) = x 3 - 6x 2 + 4x + 12, use the Intermediate Value Theorem to determine which interval(s) must contain a zero of f. (Work not required.) a. (-4,-2) yes or no b. (-2,0) yes or no c. (0,2) yes or no d. (2,4) yes or no 4. (8 points) Solve the equation. Check all proposed solutions. Show work in solving and in checking, state your final conclusion. X ( X − 2 ) + − 8 X 2 − 4 = 0 ; Multiply left by (x+2) to get the same denominator on the bottom. ( X + 2 ) X ( X + 2 ) ( X − 2 ) + − 8 X 2 − 4 = 0 X 2 + 2 X X 2 − 4 + − 8 X 2 − 4 = 0

X 2 + 2 X − 8 X 2 − 4 = 0 ( X + 4 )( X − 2 ) ( X + 2 )( X − 2 ) = 0 X + 4 X + 2 = 0 Set numerator equal to 0 and solve X + 4 = 0 X = -4 Show work to check X X − 2 + − 8 x 2 − 4 = 0 If X = -4 − 4 − 4 − 2 + − 8 (− 4 ) 2 − 4 = 0 4 6 + − 8 16 − 4 = 0 4 6 + − 8 12 = 0 4 6 + − 4 6 = 0 5. (4 points) Translate this sentence about area into a mathematical equation. The distance a dropped item falls is inversely proportional to the square of the time it falls. Let d stand for distance and t stand for time.) d = 1 t 2 6. (8 points) Look at the graph of the quadratic function and complete the table. (no explanation required) Graph Fill in the blanks Equation

In Quiz 2, problem #10, we saw that the profit function for this scenario is P(x) = -.03x 2 + 24.75x - 3200 a. The profit function is a quadratic function and its graph is a parabola. Does the parabola open up or down? Down because the leading coefficient is negative. b. Find the vertex of the profit function P(x) using algebra. Round x to nearest integer. Show algebraic work. P(x) = -.03x 2 + 24.75x – 3200 Knowing that it is the y = ax 2 + bx + c form, then the x-coordinate of the vertex is – b 2 a x-coordinate= – b 2 a = − 24.75 2 ( − .03 ) = 412.5 The substitute the x-coordinate into the original equation to solve for P(x) or y. P(412.5) = -.03(412.5) 2 + 24.75(412.5) – 3200 P(412.5) = -5104.6875 + 10209.375 – 3200 P(412.5)= 1904.6875 c. State the maximum profit and the number of widgets which yield that maximum profit: The maximum profit is $1,904.69 ($1,904.68) when 412.5 (413 because you cannot sell half a widget) widgets are produced and sold. d. Determine the price to charge per widget in order to maximize profit. p(x) = 30 – 0.03x p(412.5) = 30 - 0.03(412.5) p(412.5)= 17.625 The widget needs to be sold at $17.63 to maximize profit. 9. (8 points) Let f(x) = x 2 2 − x 2 2 x − − 8 15 (no explanation required) a. State the y-intercept (-∞,∞) b. State the x-intercept(s) (-∞,∞) c. State the vertical asymptotes X = -3 and 5 d. State the horizontal asymptote Y = 2 10. (8 points) Consider the equation 3x 2 + 9 = 11x . Use the quadratic formula to find the complex solutions (real and non-real) of the equation, and simplify as much as possible. Show work. 3x 2 -11x+9 = 0

x = −(− 11 ) ± √ (− 11 ) 2 − 4 ( 3 )( 9 ) 2 ( 3 ) x = 11 ± √ 121 − 108 6 x = 11 ± √ 13 6 x = 11 − √ 13 6 and x = 11 √ 13 6 11. (8 points) Rationalize and simplify the complex expression as much as possible, writing the result in the form a + bi, where a and b are real numbers. Show work. 3 8 − + 4 i i 8 + i 3 − 4 i multiply of complex conjugate 8 + i 3 − 4 i * 3 + 4 i 3 + 4 i Foil numerator and denominator = 24 + 32 i + 3 i + 4 i 2 9 + 12 i − 12 i − 16 i 2 Where i 2 is -1 = 24 + 35 i + 4 (− 1 ) 2 9 − 16 (− 1 ) 2 = 24 + 35 i + 4 9 + 16 = 20 + 35 i 25 Simplify = 4 + 7 i 5 = 4 5 + 7 5 i 12. (15 points) Let f(x) = -x 3 - 3x 2 + x + 3. When factored, f(x) = -(x-1)(x+3)(x+1) a. Which scenario illustrates the end behavior of the polynomial function? i. Up right, Up left ii. Up right, Down left iii. Down right, Down left iv. Down right, Up left b. State the y-intercept (-∞,∞) c. State the real zeros: X = 1, x = -3, x = -1 d. State which graph below is the graph of f(x) . D.

List any vertical asymptotes.: X = 0 and x = -5 List any horizontal asymptotes: Y = 0 Select the appropriate function. a. f(x) = x 2 x + 2 5 x b. f(x) = x 2 −5 7 x c. f(x) = x 2 −7 5 x d. f(x) = 7 X 2 + 5 X