First Semester Exam Review Answers, Hints and Solutions In the many pages that follow, you will find answers, solutions and hints to the first semester exam review problems. If you still have questions about any of the problems after looking over this guide, be sure to come see me for help. Remember... if my lights are on, but my door is closed I am in the building... just hang out and I will be back! Feel free to e mail me questions during break, I will respond to you as soon as I can. You can also email me over the weekend before exams and I will get back to you as soon as possible. 1) a) In your own words, explain what a function is. answers will vary b) In your own words, explain the meanings of domain and range. answers will vary c) For each scenario below, determine whether the statements use the word function in ways that are mathematically correct. If so, what would their domain and range be? The sales tax on a purchased item is a function of the selling price. Yes. The amount that you pay in sales tax will increase as the price of the item purchased increases. Your score on this pre-calculus exam is a function of the number of hours you study the night before the exam. No. The length of time that you study the night before an exam does not necessarily determine your score on the exam. The amount in your savings account is a function of your salary. No. During the course of a year, for example, your salary may remain constant while your savings account balance may vary. That is, there may be two or more outputs The speed at which a free-falling baseball strikes the ground is a function of the height from which it was dropped. Yes. The greater the height from which the ball is dropped, the greater the speed with which the ball will strike the ground. a) Write a mathematically correct scenario where the domain is [0,24] and the range is [28, 50] Scenarios will vary.
2) Use the information in the table to answer each question. a) What are the three real zeros of the polynomial function f? zeros of f(x) = -2, 1, 4 b) What can be said about the behavior of the graph of f at x = 1? The graph touches (or bounces) the x-axis at x = 1 c) What is the least possible degree of f? Explain. The least possible degree of the function is 4. Explanations will vary. Interval Value of f(x) Positive ( 2, 1) Negative (1, 4) Negative Positive d) Can the degree of f ever be odd? Explain? No - explanations will vary e) Is the leading coefficient of f positive or negative? Explain. Yes - explanations will vary f) Write an equation for f. (There are many correct answers!) Equations can vary Hint: leave your equation in factored form and use the least possible degree. g) Sketch a graph of the equation you wrote in part (f). Graph depends on equation from part f. 3) The graph of one of the following functions is shown below. Identify the function shown in the graph. Explain why each of the others is not the correct function. Use a graphing utility to verify your result. - do the graphing on your own, graphs not shown in this answer key. a) Cannot have this graph since it also has a zero at x = 0 b) Cannot have this graph since it is a quadratic function. It's graph is a parabola c) This is the correct function. It has two real zeros, x = 2 and x = 3.5, and it has a degree of four, needed to yield three turning points. d) Cannot have this graph since it also has a zero at x = 1. In addition, since it's only of degree three, it would have at most two turning points.
4) Sketch a graph of the function given by Sketch graph on your own. Graph not shown in this solution guide. Explain how the graph of each function g differs from the graph of f. Determine whether g is odd, even or neither. a) b) Up 2, Even Left 2, Neither c) d) Flip over y axis, Even Flip over x axis, Even e) f) Horizontal Stretch (wider or pull to x axis), Even Vertical Shrink (thinner or pull to y axis), Even g) h) No transformation from f, Neither Composition of f to f, Even 5) Find formulas for the following functions: a) b)
6) a) State the analytic test for symmetry with respect to the x-axis and use the test to show that is symmetric with respect to the x-axis. To reflect over the x axis, (x, y) (x, y) To use the test, put in (x, y) and show how the equation simplifies back to it's original form. b) State the analytic test for symmetry with respect to the y-axis and use the test to show that is symmetric with respect to the y-axis. To reflect over the y axis, (x, y) ( x, y) To use the test, put in ( x, y) and show how the equation simplifies back to it's original form. c) State the analytic test for symmetry with respect to the origin and use the test to show that is symmetric with respect to the origin. To reflect over the origin, (x, y) ( x, y) To use the test, put in ( x, y) and show how the equation simplifies back to it's original form. d) 10) Complete the definition: The function f is an even function if and only if Use your definition to show that is an even function.... f is symmetric to the y axis. (A polynomial will have all even exponents) So, to reflex over the y axis: (x, y) ( x, y) To show the symmetry, show how the equation as all even exponents (if the equation is a polynomial) and/ or how when ( x, y) is put into the equation, it will simplify back to it's original form. e) Complete the definition: The function f is an odd function if and only if Use your definition to show that is an odd function.... f is symmetric to the origin. (A polynomial will have all odd exponents) So, to reflex over the origin: (x, y) ( x, y) To show the symmetry, show how the equation as all odd exponents (if the equation is a polynomial) and/ or how when ( x, y) is put into the equation, it will simplify back to it's original form. 7) Given a fourth degree polynomial equation and two roots, describe how you would find the remaining two roots. Use both mathematical notation as well as a written description. Begin with the following problem: with roots of -1 and 2. (be sure to show all work and fill in the details missing in the explanation below.) Since 2 is a root, (x 2) is a factor. Since 1 is a root, (x + 1) is a factor. Using long division or synthetic substitution, we can find that: (x 2)(x + 1)(x 2 x +2) is the factored form of the original polynomial. To find the remaining factors use the quadratic formula, or completing the square or polysmlt on your calculator and factor x 2 x + 2. 8) When will the graph of a fourth degree polynomial have a W-shape or an M-shape? Generalize about all polynomials of even degree. The sign of the leading coefficient determines the shape. If it is positive, the graph will have a "w shape". If it is negative, the graph will have an "m shape". (the graph can be "u shaped", depending on whether some of the coefficients are zero. example: x 2 ) For all polynomials of even degree, the graph is either positive at both the left and right ends if the leading coefficient is positive or negative at both the left and right ends if the leading coefficient is negative. a > 0 a < 0
9) Consider the polynomial function: f(x) = x 4-8x 3 + 21x 2-8x + 20. a) According to the Rational Root Theorem, what are the possible rational zeros of this polynomial? (just list them don t find them!) b) Which of the possible rational zeros are, in fact, zeros of f? (the calculator is your friend here!) c) Factor this polynomial completely. a) ±1, ±2, ±4, ±5, ±10, ±20 b) This polynomial has no real zero. c) (x - i)(x + i)[x - (4 + 2i)][x - (4-2i)] Simplify to get: 10) a) In your own words, explain how to form the rows of Pascal s Triangle. Answers depend on your response. b) Form rows 8-10 of Pascal s Triangle. Hint: can you remember a way to do this without writing all 10 rows? 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1 c) How many terms are in the expansion of? n + 1 d) How do the expansions of and differ? signs are the same in and signs alternate in
11) Consider the general equation of a conic written in the form: a. Explain how you can tell if the equation is an equation of a circle. Write an equation in general form of a circle whose center is not the origin. Graph the equation. b. Explain how you can tell if the equation is an equation of a ellipse. Write an equation in general form of a ellipse whose center is at the origin. Graph the equation. c. Explain how you can tell if the equation is an equation of a parabola. Write an equation in general form of a parabola with vertex at (-1, 2). Graph the equation. d. Explain how you can tell if the equation is an equation of a hyperbola. Write an equation in general form of a hyperbola with a vertical asymptote on the y-axis. Graph the equation. a) If A = C. When you write your equation, first write it in standard form (center radius form) and then change it into general form by multiplying it out. Use this same method for parts b d b) If AC>0 or If A C and A and C have same signs. c) If AC=0 or If A or C = 0 or If the A or C term is missing. d) If AC<0 of If A and C have opposite signs. *Be sure you can graph each part above! The graphs are not included in this solution guide!* 12) A tour boat travels between two islands that are 12 miles apart (see figure below). For a trip between the islands, there is enough fuel for a 20-mile trip. a) Explain why the region in which the boat can travel is bounded by an ellipse. Since by definition of locus of points of an ellipse, the outer bound the boat can travel is an ellipse. The Islands are the foci. b) Let (0, 0) represent the center of the ellipse. Find the coordinates of each island. I1: (-6, 0) I2: (6, 0) c) The boat travels from one island, straight past the other island to the vertex of the ellipse, and back to the second island. How many miles does the boat travel? Use your answer to find the coordinates of the vertex. 20 miles; V(10, 0) d) Use the results from parts (b) and (c) to write an equation for the ellipse that bounds the region in which the boat can travel.
13) Graph the functions given by and and use the graph to solve each inequality. a) b) 10 x < e x for all x < 0 10 x > e x for all x >0 At x = 0, 10 x and e x are equal. 14) Dr. Tricia MacMillan has a problem. Every day she leaves her apartment in London at the crack of dawn and heads for Milliway s, where she purchases a delicious cup of piping hot coffee. She drinks this coffee while walking to her office. The problem is that sometimes she burns her tongue badly with her first sip, while other times she waits too long and her coffee gets cold. The latter case is the worst, because besides doing a pretty bad job of keeping you warm, cold coffee tastes terrible. As it drops below a certain temperature, coffee undergoes a chemical reaction which turns even the most expensive brand into something that tastes absolutely filthy. Being a mathematician, Dr. McMillan doesn t just get mad, she gets more coffee and does an experiment. She wants to figure out exactly when she can take her first sip without burning herself, and from that point, how much time she has before the coffee turns bad. Every one of her mornings for the next week is spent in Milliway s with an over thermometer and a cup of fresh coffee. After much painful experimentation, Dr. MacMillan determines that if the temperature of the coffee is above 140 0 F, it burns her tongue. If the temperature drops below 105 o F, the coffee undergoes the reaction and becomes undrinkable (unless she s already burnt her tongue so badly in the first experiment that she can t taste a thing). Just like every other substance in the universe, coffee obeys Newton s Law of Cooling. It s temperature as a function of time is given by: T(t) = T s + D o e -kt Note that there are three parameters in this equation. One is the outside temperature, and one depends on the initial temperature of the coffee. For a typical Styrofoam cup, k = 0.05, if t is measured in minutes. a) Why is this constant positive? THE COFFEE COOLS DOWN OVER TIME; IT DOES NOT WARM UP Dr. MacMillan scoffs at Styrofoam. She is the proud owner of a Sirius Cybernetics corporation thermos (only 35% asbestos!). For this thermos the constant is K = 0.03. b) Which does a better job of keeping the coffee warm, the Styrofoam cup or the thermos? How does knowing the value of k allow you to figure out the answer? THE THERMOS IS BETTER. A SMALLER VALUE OF K CORRESPOND TO A SMALLER RATE OF CHANGE OF THE COFFEE'S TEMPERATURE. The next day, Dr. MacMillan leaves Milliway s with a thermos full of coffee at 160 o F. It is 8:30 AM, and the outside temperature is 42 O. c) Find Ts and Do and rewrite with the appropriate constants for this situation. (Let the time, t, be measured in minutes and ltet t=0 stand for 8:30 AM) d) How long must she wait before she is able to drink the coffee? e) At what time will the coffee fall below 105 o and become undrinkable? f) How much time does Dr. MacMillan have to drink her coffee?
15) a) Solve the equation Explain each step. Be sure to explain each step. To finish solving for x, use quadratic formula or completing the square or polysmlt on your calculator. b) Solve the equation Explain each step. Method 1: Reverse Foil... Method 2: Substitution then Reverse Foil... Be sure to explain each step no matter which method you choose. 16) Graph the following pairs of curves, then find each pair s point(s) of intersection. a) b) c) d) (find a and b that make these curves intersect at the point (1, 2)). You will need to adjust your viewing window in order to find all the solutions! a) (1, 3), ( 2, 6), ( 3, 21) b) (2, 4.693) c) No intersections d) Be sure to show work! a =1, b = 1
17) You have two options for investing $500. The first earns 7% compounded annually and the second earns 7% simple interest. The figure shows the growth of each investment over a 30 year period. a) Identify which graph represents each type of investment. Explain your reasoning. Curve 1: Annually, exponential growth Curve 2: Simple interest, linear growth b) Verify your answer from part (a) by finding the equations that model the investment growth and graphing models. c) Which option would you choose? Explain your reasoning. Answers and explanations will vary. 18) 25) A lab culture initially contains 500 bacteria. Two hours later, the number of bacteria has decreased to 200. Find the exponential decay model of the form: that can be used to approximate the number of bacteria after t hours.
19) A children s play area is being built next to a circular fountain in the park. A fence will be erected around the play are for safety. A diagram of the area is shown below. a) How long will the fence need to be in order to enclose the area? (Show work!) about 140.7 feet b) The park commission is planning to enlarge the play area. Do you think they should enlarge it to the east or the west? Why? answers vary 20) Every year the Centerville Chamber of Commerce sponsors a Chat-a-thon where happy couples stand four feet apart and compete to see who can spend the most time talking about their relationship. There a two ways that couples tend to lose - they walk away to see what is on television, or they fall asleep standi up. It s actually quite an adorable sight to see two people in love, one six feet tall, the other five feet tall standing four feet apart, leaning against each other in sleep: How far off the ground are the tops of their heads? The height is about 4.96 feet.