Math 1314 Lab Module 2 Name _____________________ A. Quadratic Equations and Functions Show all work! Simplify and clearly indicate all answers. 1. For , answer the following questions. a. Find the values of the x-intercepts. b. State the coordinates of the y-intercept. c. State the vertex. d. State the axis of symmetry. e. State the direction of the parabola. f. Decide whether there is a relative maximum or minimum, then state it. g. Graph the function accurately using the above information. Math 1314 2. For Lab Module 2 , answer the following questions. a. Find the values of the x-intercepts. b. State the coordinates of the y-intercept. c. State the vertex. d. State the axis of symmetry. e. State the direction of the parabola. f. Decide whether there is a relative maximum or minimum, then state it. g. Graph the function accurately using the above information. h. State the domain. i. State the range. j. State the interval of x where the graph is increasing. k. State the interval of x where the graph is decreasing. 2 Math 1314 Lab Module 2 B. Applications of Quadratic Functions 1. Laura owns and operates Aunt Linda's Pecan Pies. She has learned that her profits, the sale of x cases of pies, are given by - . a. The company will \"break-even\" when the profit is zero. How many cases of pies should Laura sell in order to break-even? (Solve for x when .) b. How many cases of pies should she sell in order to maximize profit? c. What is the maximum profit? 2. John wants to build a corral next to his barn. He has 300 feet of fencing to enclose three sides of his rectangular yard. a. What is the largest area that can be enclosed? b. What dimensions will result in the largest yard? 3 , from Math 1314 Lab Module 2 3. The function models the number (in thousands), murders committed in a certain state in the United States years after 1960. Let represent 1960, represent 1961, and so on. 4 , of a. Determine the year in which the most violent crimes were committed. b. Approximately how many violent crimes were committed during this year? c. Using a graphing calculator, graph . Were the number of violent crimes increasing or decreasing during the years 2000 to 2005? 4. A ball is thrown vertically upward from the top of a building 1600 feet tall with an initial velocity of 80 feet per second. The distance , in feet, of the ball from the ground after t seconds is . a. After how many seconds is the ball at its maximum height? b. Find the maximum height of the ball. c. How long would it take the ball to hit the ground? Math 1314 Lab Module 2 C. Finding Real Zeros of Polynomial Functions 1. For a. Find 5 , using synthetic division. b. Is 10 a zero of the function? Explain. c. Use synthetic division to determine if is a factor of . d. Is -1 a zero of the function? Explain. e. List all of the zeros of the polynomial. f. Write the polynomial function as a product of linear factors. 2. For the function , a. State the degree of the polynomial. b. Use the Rational Zero Theorem to list all of the possible rational zeros. c. Use a graphing calculator to determine which numbers in the list of possible rational zeros are probable rational zeros (indicated by the x-intercepts of the graph). Math 1314 Lab Module 2 d. Use synthetic division to determine one rational zero. e. Use synthetic division or other algebraic methods to find all remaining zeros. List all of the zeros of the polynomial function. f. Write the polynomial function as a product of linear factors. D. Finding Complex Zeros of Polynomial Functions 1. For the function a. State the degree of the polynomial b. State the number of zeros the polynomial function will have. c. Use the Rational Zero Theorem to list all of the possible rational zeros. d. Use your calculator to determine which numbers in the list of rational zeros are probable rational zeros. e. Use synthetic division to verify one rational zero. 6 Math 1314 Lab Module 2 f. Use synthetic division or other algebraic methods to find all remaining zeros. List all of the zeros of the polynomial function. 7 2. For the function a. State the degree of the polynomial b. State the number of zeros the polynomial function will have. c. Given that is a zero, find all remaining zeros. List all of the zeros of the polynomial function. 3. Given a cubic polynomial function following questions. Justify each answer. a. How many x-intercepts can there be? b. Does the degree of this polynomial function guarantee any x-intercepts? c. Will the graph pass through the origin? d. Could the graph \"touch\" the x-axis in two different places? , answer the Math 1314 Lab Module 2 e. Identify the end behavior of the graph. 8 f. If it is known that one zero is real and another zero is imaginary, what can be determined about the remaining zeros? E. Rational Functions 1. A rare species of insect was discovered in the rain forest. In order to protect the species, environmentalists declare the insect endangered and transplant the insects into a protected area. The population of the insect months after being transplanted is given by . a. How many insects were discovered? In other words, what was the population when ? b. What will the population be after 5 years? c. Determine the horizontal asymptote of area can sustain? d. Graph . . What is the largest population that the protected Math 1314 Lab Module 2 9 2. Based on data from the U.S. Department of Agriculture, the average number of acres per farm x years after 2000 can be approximated by the model . a. Use the model to estimate the average number of acres per farm in 2005. b. Use the model to predict the average number of acres per farm in 2012. c. Find and interpret the zero of the rational function. Does this result make sense within the context of the problem? 3. The function models the pH level, after a person eats food containing sugar. of the human mouth x minutes a. Determine to the nearest tenth the pH level of the human mouth 42 minutes after a person eats food containing sugar. b. What is the equation of the horizontal asymptote associated with this function? Describe what this means in terms of the mouth's pH level over time. Math 1314 Lab Module 2 10 4. Suppose a company that manufactures mountain bikes has a fixed monthly cost of $100,000 and that it costs $100 to produce each mountain bike. a. Write the cost function, C, of producing x mountain bikes. b. Write the average cost function, , of producing x mountain bikes. c. Find and interpret , , and . d. What is the horizontal asymptote for the graph of the average cost function, this represents for the company. ? Describe what