1. Write an expression that represents the width of a rectangle with length x 5 and area

Transcription

1 1. Write an expression that represents the width of a rectangle with length x 5 and area x 1x 7x 60. x 7x 1x x 7x x 5 x x x 5 x x 5. A board of length x cm was cut into two pieces. If one piece is 7 cm, express the length of x the other board as a rational expression. x ( x)( x) x ( x ) 6x ( x ) 6x ( x)( x) Revised Feb 01 Page 1

2 . Use the following expressions to answer the questions. x 5 A x x B x x x C x 7 D x (a) Explain how to add two rational expressions. Simplify A Bas an example. Be sure to give reasons for each step. Simplify completely. (b) Explain how subtracting (such as simplifying A B) would be different from adding. (c) Explain how to multiply rational expressions, simplifying BC as an example. (d) Show how to divide B by C. (e) Explain how you would solve a rational equation like C D. Then describe, in detail, the strategy for solving an equation like B C D. Do not completely solve the equations. Instead, concentrate on the first two or three steps in solving; show what to do and explain why.. The rate of heat loss from a metal object is proportional to the ratio of its surface area to its volume. (a) What is the ratio of a steel sphere s surface area to volume? (b) Compare the rate of heat loss for two steel spheres of radius meters and meters, respectively. 5. Multiply. Simplify your answer. 6 8x y 9xy z z y x y z 5 6x z 6x yz x y z Revised Feb 01 Page

3 6 8x z x z 6. Which expression represents the quotient? xz x z x z x z x z x z x z x z x z 7. Last week, Wendy jogged for a total of 10 miles and biked for a total of 10 miles. She biked at a rate that was twice as fast as her jogging rate. (a) Suppose Wendy jogs at a rate of r miles per hour. Write an expression that represents the amount of time she jogged last week and an expression that represents the amount to time she biked last week. (hint: dis tan ce rate time ) (b) Write and simplify an expression for the total amount of time Wendy jogged and biked last week. (c) Wendy jogged at a rate of 5 miles per hour. What was the total amount of time Wendy jogged and biked last week? 8. Which expression is equivalent to y 8x x y for all xy, 0? x y 5 x y y x y 9 8x Revised Feb 01 Page

4 x 9. Which set contains all the real numbers that are not part of the domain of f( x)? x x {8} {-} {-, 8} {-8, } x Solve the equation. x 9 x7 x No solution 11. A sight-seeing boat travels at an average speed of 0 miles per hour in the clam water of a large lake. The same boat is also used for sight-seeing in a nearby river. In the river, the boat travels.9 miles downstream (with the current) in the same amount of time it takes to travel 1.8 miles upstream (against the current). Find the current of the river. 1. A baseball player s batting average is found by dividing the number of hits the player has by the number of at-bats the player has. Suppose a baseball player has 5 hits and 10 at-bats. Write and solve an equation to model the number of consecutive hits the player needs in order to raise his batting average to Explain now you found your answer. 1. Which intervals correctly define the domain of 1 f( x) x (,) and (, ) (, ) and (, ) (, ) and (, ) (, ) and (, ) Revised Feb 01 Page

5 1 1. Which statement is true for the function f( x) x? is not in the range of the function. is not in the domain of the function. - is not in the range of the function. - is not in the domain of the function. 15. Convert 60to radians. radians radians 6 radians radians 16. An analog watch had been running fast and needed to be set back. In resetting the watch, the minute hand on the watch subtended an arc of 5 radians. Part A: Suppose the radius of the watch is 1 unit. What is the length of the arc on the outside of the watch that the angle subtends? Part B: If the watch was at 10:55 before being reset, what is the new time on the watch? Part A: 10 units Part B: 9:05 Part A: 5 units Part B: 10:05 Part A: 5 units Part B: 9:0 Part A: 10 units Part B: 8:0 Revised Feb 01 Page 5

6 17. Convert radians to degrees Convert 16 radians to degrees. 19. Suppose each paddle on the wall of a clothes dryer makes 80 revolutions per minute. Part A: What angle does one paddle subtend in 10 seconds? Give your answer in radians. Part B: Write an algebraic expression to determine the measure in radians of the subtended angle after x seconds. Show how the units simplify in your expression. Part C: You are interested in determining the total distance a point on the drum travels in a 0- minute drying cycle. Can you use your expression from Part B? What other information, if any, is needed? Explain. 0. What is the exact value of tan? 1. Which expression has the same value as tan? tan tan cos sin Revised Feb 01 Page 6

7 . What is the reference angle corresponding to 7? 5 7. A regular hexagon is inscribed in the unit circle. One vertex of the hexagon is at the point 1,. A diameter of the circle starts from that vertex and ends on another vertex of the hexagon. What are the coordinates of the other vertex? 1, 1, 1, 1, Revised Feb 01 Page 7

8 . For what angles x in 0, does the cos x have the same value as cos and 5 cos cos and 7 cos cos and 7 cos cos and 5 cos 5. For which radian measures x will tan x be negative? sin? ,,,,,, ,,,,,, ,,,,,, ,,,,,, The diameter of a bicycle tire is 0 in. A point on the outer edge of the tire is marked with a white dot. The tire is positioned so that the white dot is on the ground, then the bike is rolled so that the dot rotates clockwise through an angle of radians. Part A: To the nearest tenth of an inch, how high off the ground is the dot when the wheel stops? Show your work. Part B: What distance was the bicycle pushed? Round your answer to the nearest foot. Part C: Would changing the size of the tire (value of r) change either of the answers found in Parts A or B? Explain your reasoning. 7. A ribbon is tied around a bicycle tire at the standard position 0. The diameter of the wheel is 6 inches. The bike is then pushed forward 0 feet from the starting point. In what quadrant is the ribbon? Explain how you obtained your answer. Revised Feb 01 Page 8

9 8. Two friends counted evenly spaced seats on a Ferris wheel. As they boarded one of the seats, they noticed the edge of the wheel was 1 meter off the ground. They learned from the operator that the diameter of the wheel was 8 meters. After they got seated and started moving, in a counterclockwise direction, they counted 1 chairs pass the operator, and then the Ferris wheel was stopped on the fourteenth chair to load another passenger. Part A: Design a representation of the Ferris wheel and locate where the friends were when the wheel stopped to load the next passenger. Part B: How many radians had they rotated through in the time before they stopped? Part C: To the nearest tenth of a meter, how far above the ground were they? Show your work. 9. Which is the equation of the graph shown below? sin f x x sin f x x 1 f x sin x 1 f x sin x 0. The graph of which function has a period of and an amplitude of? 1 y sin x y sin x 1 1 y sin x 1 y sin x Revised Feb 01 Page 9

10 1. Which function has an amplitude of and a period of? f x cosx cos f x x 1 f x cos x 1 f x cos x. Which function has an amplitude of and a period of? f x x f x cos x 1 f x cos x f x cos x cos f x x?. Which of the following is a vertical asymptote of the graph of tan (E) x x x x x Revised Feb 01 Page 10

11 . What is the equation for the graph shown? ysin x 8 y6sin x ycos x 8 y6cos x (E) ysin x 1,1 9, 5. Which function has an amplitude of and a period of 1? y sin x y cos x y sin x 1 y cos x 6. Which function is represented by the graph shown? y tan x 1 y tan x y tan x y tan x Revised Feb 01 Page 11

12 7. Write an equation of the form y asin bx, where a 0 and b 0, with amplitude and period Write a function for the sinusoid.,8, 9. A sound wave models a sinusoidal function. Part A: If the wave reaches its maximum at,1 and its minimum at,0, what are the shift, amplitude, and period of the function? Part B: Write the function that models this sound wave. Part C: Graph the function. 0. Is the function y sin x in the form y asin bx h k? Why or why not? How does the amplitude and period of the function compare to the amplitude and period of y sin x? How does the graph of the function compare to the graph of y sin x? Revised Feb 01 Page 1

13 1. Graph y sin x. Revised Feb 01 Page 1

14 . Using f x cos x as a guide, graph g x cos shift. x. Identify the x-intercepts and phase x-intercepts: x n where n is an integer; phase shift: units to the right x-intercepts: x n where n is an integer; phase shift: units to the left x-intercepts: x n where n is an integer; phase shift: units up x-intercepts: x n where n is an integer; phase shift: units down Revised Feb 01 Page 1

15 . Find the amplitude and period of the graph of y cos x.. Find the amplitude and period of the graph of y cos6x. x 5. Graph one cycle of the graph of the function cos f x. 6. Graph one cycle of the graph of the function 6sin x f x x 7. Graph y tan. Include vertical asymptotes in your sketch. 8. The graph of a sine function has amplitude 5, period 7, and a vertical translation units down. Write an equation for the function. 9. The graph of a cosine function has amplitude, period 90, and a vertical translation units down. Write an equation for the function. Then sketch the graph without using graphing technology. Revised Feb 01 Page 15

16 50. Sketch the graphs of y cos x, y cos x, and y cos x. Tell how the graphs are alike and how they are different. 51. Consider the related equations y sin x, y sin x, and y sin x. Explain the effect that the coefficient has on the graphs of y sin xand y sin xwhen compared to the graph of y sin x. 5. Find sin cos when sin and is in Quadrant I. Revised Feb 01 Page 16

17 5. Find tan when sin cos and is in Quadrant IV The unit circle centered at the origin has a radius of 1, and the coordinates xylocate, any point on the circle. Part A: Prove that cos sin 1for representing the central angle of the arc intercepted by the point xyand, the x-axis. Part B: Does this formula work for all values for? Explain. Part C: Without using a calculator, evaluate cos if 55. Suppose cossin 1. Part A: Solve the equation for in the interval 0,. tan. Part B: Why might you have to check solutions in Part A for extraneous roots? Part C: Suppose is between 5.5 and 6 and cossin 1. Find cos and explain your reasoning For an angle, sin and. 1 Part A: Use the Pythagorean identity to find cos. Part B: If 5 sin and, does cos change? Explain. 1 Revised Feb 01 Page 17

18 57. The area A of a trapezoid is given by the formula A 1 ( b ) 1 b where b1 and b are the lengths of the two parallel sides. Solve the formula for b. A b b1 A b b1 b A b1 b A b1 58. The formula p nc e gives the profit p when a number of items n are each sold at a cost c and expenses e are subtracted. If p 750, n 000, and e 900, what is the value of c? As a furniture salesperson, Marcy gets a % commission on all her weekly sales above $5000. Which row in the table show the composite function that will determine her commission if x represents her weekly sales? Amount Eligigle for Commission Commission Composite Function f ( x) 5000 x g( x) 0.0 x f ( g( x)) f ( x) x 5000 g( x) 0.0 x f ( g( x)) f ( x) 5000 x g( x) 0.0 x g( f ( x)) f ( x) x 5000 g( x) 0.0 x g( f ( x)) Revised Feb 01 Page 18

19 60. Denise is reviewing the change in the value of an investment. Time (decades) 0 1 Value ($1000s) Which statement can Denise use to model the data? Why is this type of function a good model for the data? 1 t v ( ) ; an exponential function is a good model because the value of the investment changes by a constant amount in each time period. 1 t v ( ) ; an exponential function is a good model because the value of the investment changes by a constant factor in each time period. v t ; a linear function is a good model because the value of the investment changes by a constant amount in each time period. v t ; a linear function is a good model because the value of the investment changes by a constant factor in each time period. 61. The area of a square is given by the function A( x) x where x is the length of a side. When p p centimeters of wire are bent to create a square, the function S( p) can be used to find the length of each side of that square. Which function gives the area of a square created by bending p centimeters of wire in terms of p? Ap ( ) p A( p) p p Ap ( ) 16 p Ap ( ) Revised Feb 01 Page 19

20 Total Number of Ladybugs 6. Amy recorded the total number of ladybugs observed in a garden over a 7-day period. The scatterplot below represents the data she collected. Ladybugs Seen in Garden Over Time Number of Days Which type of function do these data points best fit? Cubic Exponential Linear Quadratic 6. Public Service Utilities uses the equation y a blog x to determine the cost of electricity where x represents the time in hours and y represents the cost. The first hour of use costs $6.66 and three hours cost $ a) Determine the value of a and b in the model. b) What is the x-intercept of the graph of the model? What is the real world meaning of the x -intercept? c) Use the model to find the cost for 65 hours of electricity use. d) If a customer can afford $0 per month for electricity, how long can he or she have the electricity turned on? Revised Feb 01 Page 0

21 6. On an xy - coordinate plane the earth is located at (, -1) and an asteroid is traveling on the path of x e. g( x) a) Write an equation representing the distance from the earth to the asteroid. 8 b) If the asteroid is currently located at (, e ), what is the distance from the earth to the asteroid? c) Sketch a graph of gx. ( ) d) Find the point when the asteroid is closest to the earth. 65. The relationship between the weight of a whale in tons, W, and the length in feet, L, is given by.18 W L. Which expression below would be used to find the length of a whale that weighs 50 tons? Rashid is in Biology class and has gathered data on fruit flies. The table below shows the number of fruit flies in his sample at the end of each day for a week. Day 1 5 # of flies If the population continues to grow in this manner, which function will Rashid use to predict the population of fruit flies on any given day? p( t) 5t p( t) 5( x ) 0 p( t) ln t 1 pt ( ) 5 t Revised Feb 01 Page 1

22 67. The table gives the number of inner tubes, I, sold in a bike shop between 1985 and Determine which model best fits the data. Year ( t) Inner tubes ( I) Linear Absolute value Quadratic Exponential 68. Which function best fits the data shown in this scatter plot? f ( x) x f ( x) x ( ) x f x 1 f ( x) x Revised Feb 01 Page

23 69. This graph shows the scatter plot of a data set and a curve of fit. What is the residual plot for this data set and curve? Revised Feb 01 Page

24 70. Which residual plot show the best fit between a function and a data set? Revised Feb 01 Page

25 71. This residual plot shows the relationship between a linear function and a data set. Is it possible that this function is a line of best fit for the data? How can you tell? The function could be a line of best fit for the data, because the absolute value of each residual is or less. The function could be a line of best fit for the data, because the residuals are close to linear. The function is not a line of best fit for the data, because the residuals form a linear pattern that does not have a slope near zero. The function is not a line of best fit for the data, because the residuals include both and The graph represents a residual plot for a data set and a linear model. Based on the residual plot, discuss the goodness of fit of the linear model. Revised Feb 01 Page 5

26 7. Create a residual plot for the table using the linear model y1.8x 8. Age, x (months) Baby's Weight, y (pounds) The graph below shows the change in temperature of a burning house over time. a) Describe the graph. b) This graph was found in an old math book and next to it was written: Rise of temperature = t 0.5 Show that this function does not describe the graph correctly. c) Assume that the power function 0.5 r At is a good description of the graph. Find a reasonable value for A. Graph the new function. d) Compare the graph in part (c) to the original one. Do you think that a different power of t might result in a better model? Would a larger or smaller power produce a better fit? Explain. e) Use the original graph to find data. Carry out a power regression on the data to find a function that would produce a better fit. Revised Feb 01 Page 6

27 75. Psychologists try to predict the activation of memory when a person is tested on a list of words they learned. The following model is used to make this prediction: A ln( n) 0.5ln( T) 0.5ln( L) where A is the number of words learned, n is the number of exercises, T is the amount of time between learning and testing and L is the length of the list that was tested. a) Write the formula as the ln of a single expression. b) Discuss the influence on A (going up or down) when increasing n, T, and L, according to the formula. Do these results make sense? c) If you want A to be bigger than 0, what conditions must be placed on L, T, and n? 76. In Tanzania, an American scientist studied the hunting habits of chimps. The success rate of a hunt is related to the size of the hunting group. The graph below taken from American Scientist (May/June 1995), show the data that was gathered about the percentage of successful hunts for different group sizes. For the relationship between the two quantities, a regression curve was used. a) What kind of regression was used? Use this regression curve to estimate the number of members needed to be certain that the hunting will be successful. b) Looking at the data, you might think that a quadratic fit would be better than the one used. Explain why. c) What reason could you give to support a non-linear fit? Revised Feb 01 Page 7

28 77. Tides can be modeled by periodic functions. Suppose high tide at the city dock occurs at : AM at a depth of 5 meters and low tide occurs at 9:16 AM at a depth of 9 meters. Write an equation that models the depth of the water as a function of time after midnight. When will the next high and low tides occur? 78. A Ferris wheel with a radius of 5 feet is rotating at a rate of revolutions per minute. When t 0, a chair starts at the lowest point on the wheel, which is 5 feet above ground. Write a model for the height h (in feet) of the chair as a function of the time t (in seconds). 79. Storm surge from a hurricane causes a large sinusoidal wave pattern to develop near the shore. The highest wave reached the top of a wall 0 feet above sea level. The low point immediately behind this wave was 6 feet below sea level, and was 0 feet behind the peak. What is the amplitude of the sinusoid? What is the vertical shift of the sinusoid from a wave at ground level? Revised Feb 01 Page 8

29 80. The graph below shows how the reproductive rate of rodents varies depending on the season. On the x-axis, the months are grouped by season and on the y-axis, the reproductive rate is represented on a scale from poor to good. a) When is the reproduction of the rodents at the lowest? When is it at the highest? b) Put 0. and as the minimum and maximum values on the y-axis. Design a formula that describes the graph. Explain how you determined your formula. c) Suppose the reproductive rate were put on a scale from 0, for extremely poor, to 10 for extremely good. In this case, use 0 and 10 as the minimum and maximum values on the y-axis. Design a formula that describes the corresponding graph. What changes did you have to make to your formula form part (b)? Revised Feb 01 Page 9

30 81. An oscilloscope is a machine that measures the magnitude of fluctuating voltages by displaying a graph of the voltage over time. The figure below shows the shape of the fluctuating voltage. The horizontal axis displays time, t, and the vertical axis shows voltage. The person who is working with scope can see from the buttons that in this case, one step on the horizontal-axis scale is 0.5 sec and one step on the vertical-axis scale is 0. V. Design the formula for this fluctuating voltage. Explain how you determined your formula. 8. One modern application of the addition and subtraction of functions appears in the field of audio engineering. To help understand the idea behind this use, consider the following simplified example. Suppose that the function f(x) = sin(x) represents the sound of music to which you are listening. Unfortunately, there is background noise. Let g(x) = 0.75sin(1.5x) represent that noise. a) Sketch a graph of f and the sum f + g, on a single set of axes. Your graph represents both what you want to hear and what you actually do hear. They clearly are not the same. b) Now apply some mathematics to engineer away the noise. You can add a microphone to your headphones. In turn, the microphone picks up the background noise, gx, ( ) then plays back through our headphones an altered version, hx ( ). Thus what you now hear in the headphones is the sum of three functions: f (what you want to hear), g (the noise you don t want), and h (the correction for the noise). Suppose the compensating function is h( x) 0.75sin(1.5 x ). Graph the new sum, f g h, to see what you hear now. Explain the idea behind the adjustment. c) Another engineer suggests adding h( x) 0.75sin(1.5 x) instead of the h defined in part (b). Discuss the merits of that solution. Revised Feb 01 Page 0

31 8. The price of oranges fluctuates, depending on the season. The average price during a complete year is $.5 per kilogram (about. pounds). The lowest price will be paid in mid-february ($1.60 per kilo); the highest price is paid in mid-august. Assume the price fluctuations are sinusoidal in nature. a) Design a formula using the cosine to describe the price of oranges by months during a year. Let t 0represent January 1, t 1, February, and so forth. Explain how you determined your answer. b) How would your formula change if you used the sine function instead of cosine? Explain how you arrived at your answer. c) Sketch the graph of your model from part (a). Then read from your graph the times of the year when the price of the oranges will be below $ A satellite is circling west-to-east around the earth. Below are the projections of three orbits of the satellite, labeled as curves 1,, and. The projections are given in terms of longitude and latitude readings (both are in degrees). Curve 1, can be expresses as a sinusoidal function of the form y Asin( B( x C)) D. a) Determine values for A, B, C, and D to produce a sinusoidal model that you think best describes Curve 1. b) What constants in your sinusoidal model for (a) will you need to modify in order to describe Curves and? What are your models describing these curves? Revised Feb 01 Page 1