College Algebra
MAT111A - College Algebra Quiz 4 (15 Questions) The following questions were adapted from Larson, R. (2018). College Algebra (10th ed). NOTE: Be sure to show your work for the solution to each question. Partial credit may be given, even if the final answer is not correct, as long at the proper concepts are being depicted. 1. (2 Points)From the graph of a quadratic function given below, select the function’s equation from the following options. a. ?(?) = (? + 1)2 − 1 b. ?(?) = (? + 1)2 + 1 c. ℎ(?) = (? − 1)2 + 1 d. ?(?) = (? − 1)2 − 1 2. (2 Points)Find the coordinates of the vertex for the parabola defined by the quadratic function ?(?) = 3?2 − 12? + 1 a. (-12,3) b. (8,-5) c. (2,-11) d. (4,1) 3. (3 Points)For the quadratic function ?(?) = ?2 − 2? − 3 a. Use the vertex and intercepts to sketch the graph of this quadratic function. b. Give the equation of the parabola’s axis of symmetry. c. Use the graph to determine the function’s domain and range. Using the coordinate template below draw the graph of ?(?). 2 4 -2 -4 -6 6 x 2 4 -2 -4 6 -6 8 10 -8 -10 8 10 -8 -10 y 4. (3 Points)Among all pairs of numbers whose sum is 16, find a pair whose product is as large as possible. What is the maximum product? Write a formula for solving this question and determine the maximum product. 5. (2 Points)What is the degree of the polynomial ?(?) = 7?2 + 9?4 a. 2 b. 4 c. 6 d. 8 6. (2 Points)Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function ?(?) = 11?3 − 6?2 + ? + 3 a. The graph of f(x) falls the left and rises to the right b. The graph of f(x) falls the left and falls to the right c. The graph of f(x) rises the left and rises to the right d. The graph of f(x) rises the left and falls to the right 7. (4 Points)For the polynomial function ?(?) = ?4 − 6?3 + 9?2 a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. Using the coordinate template below draw the graph of ?(?). 2 4 -2 -4 -6 6 x 2 4 -2 -4 6 -6 8 10 -8 -10 8 10 -8 -10 y 8. (7 Points)The graph below shows the average price per gallon of gasoline in the United States in January for the period from 2005 through 2011. a. For which years was the average price per gallon in January increasing? b. For which years was the average price per gallon in January decreasing? c. How many turning points (from increasing to decreasing or from decreasing to increasing) does the graph have for the period shown? d. Suppose that a polynomial function is used to model the data displayed by the graph using (number of years after 2005, average January price per gallon). Use the number of turning points to determine the degree of the polynomial function of best fit. e. For the model in part (d), should the leading coefficient of the polynomial function be positive or negative? Explain your answer. f. Use the graph to estimate the maximum average January price per gallon. In which year did this occur? g. Use the graph to estimate the minimum average January price per gallon. In which year did this occur? 9. (2 Points)Perform the steps for dividing the two polynomials using long division. State the quotient, q(x), and the remainder, r(x). 2?5 − 8?4 + 2?3 + 2?2 2?3 + 1 10. (2 Points)Perform the steps for dividing the two polynomials using synthetic division. State the quotient, q(x), and the remainder, r(x). (6?5 − 2?3 + 4?2 − 3? + 1) ÷ (? − 2) 11. (2 Points)Perform the steps for the following. a. Use synthetic division to show that 3 is a solution of the polynomial equation 14?3 − 17?2 − 16? − 177 = 0 b. Use the solution from part (a) to solve this problem. The number of eggs, f(x), in a female moth is a function of her abdominal width, x, in millimeters, modeled by ?(?) = 14?3 − 17?2 − 16? + 34 What is the abdominal width when there are 211 eggs? 12. (2 Points)Refer to the graph of the rational function below and complete the statements that follow. a. As x→ -3-, f(x) → _____ b. As x → -3+, f(x) → _____ c. As x → 1-, f(x) → _____ d. As x → 1+, f(x) → _____ e. As x → -∞, f(x) → _____ f. As x → ∞, f(x) → _____ 13. (2 Points)Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of the rational function ℎ(?) = ?+7 ?2+4?−21 . Perform the steps to complete this. 14. (2 Points)Find the horizontal asymptote, if there is one, of the graph of the rational function ℎ(?) = 12?3 3?2+1 Perform the steps to complete this 15. (3 Points)Use transformations of ?(?) = 1 ? or ?(?) = 1 ?2 to graph the rational function ℎ(?) = 1 (?−3)2 + 2 Use the coordinate template below to draw the graph of ℎ(?). 2 4 -2 -4 -6 6 x 2 4 -2 -4 6 -6 8 10 -8 -10 8 10 -8 -10 y MAT111A - College Algebra Homework Problems 4 (15 Questions) The following questions were adapted from Larson, R.(2018). College Algebra (10th ed) . NOTE: Be sure to show your work for the solution to each question. Partial credit may be given, even if the final answer is not correct, as long at the proper concepts are being depicted. 1. (1 Point) From the graph of the quadratic function shown below, select the function’s equation from the available options. a. ?(?) = ?2 + 2? + 1 b. ?(?) = ?2 − 2? + 1 c. ℎ(?) = ?2 − 1 d. ?(?) = −?2 − 1 2. (1 Point) Find the coordinates of the vertex for the parabola defined by the quadratic function ?(?) = 2?2 − 8? + 3 The vertex is at x = _____ and y = _____ 3. (2 Points) For the quadratic function ?(?) = 4 − (? − 1)2 a. Use the vertex and intercepts to sketch the graph of this quadratic function. b. Give the equation of the parabola’s axis of symmetry. c. Use the graph to determine the function’s domain and range. Use the coordinate template below to draw the graph 2 4 -2 -4 -6 6 x 2 4 -2 -4 6 -6 8 10 -8 -10 8 10 -8 -10 y 4. (2 Points) A ball is thrown upward and outward from a height of 6 feet. The height of the ball, f(x), in feet, can be modeled by ?(?) = −0.8?2 + 2.4? + 6 , where x is the ball’s horizontal distance, in feet, from where it was thrown. a. What is the maximum height of the ball and how far from where it was thrown does this occur? b. How far does the ball travel horizontally before hitting the ground? Round to the nearest tenth of foot. c. Graph the function that models the ball’s parabolic path. Use the coordinate template below to draw the graph. 2 4 -2 -4 -6 6 x 2 4 -2 -4 6 -6 8 10 -8 -10 8 10 -8 -10 y 5. (1 Point) Which function is a polynomial function with degree 5? a. ?(?) = 5?2 + 6?3 b. ?(?) = 7?2 − 2?5 + 1 c. ?(?) = 7?5 + 2?2 + 1 ? d. ?(?) = ?2 + 2? + 5 6. (1 Point) Using the Leading Coefficient Test, what is the end behavior of the graph of the polynomial function f(?) = −5?4 + 7?2 − ? + 9 ? a. The graph of f(x) falls the left and rises to the right b. The graph of f(x) falls the left and falls to the right c. The graph of f(x) rises the left and rises to the right d. The graph of f(x) rises the left and falls to the right 7. (2 Points) For the polynomial function ?(?) = −?4 + 4?2 a. Use the Leading Coefficient Test to determine the graph’s end behavior. b. Find the x-intercepts. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept. c. Find the y-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points