Name: Practice Integrated II Final Exam Review The questions below represent the types of questions you will see on your final exam. Your final will be all multiple choice however, so if you are able to answer the questions below you will be just fine. EXPONENTIAL FUNCTION (UNIT 1) 1. Suppose that x 3 = 64 and x 5 = 1,024 and (64)(1,024) = x n. Find the value of n. Explain your reasoning. 2. Are 3(4 x ) and 12 x equivalent for all x? Explain your reasoning. 3. Darius has just been hired as an apprentice electrician. His starting wage is $9 per hour. He has been told that if he gets an excellent performance evaluation each year he will get a 6% raise each year until he reaches a wage of $25 per hour. Assume that Darius always gets an excellent performance evaluation for his work. a. Will Darius s wage increase by the same dollar amount each year? Explain your reasoning. b. What is Darius s wage after two years? Explain your reasoning or show your work. c. Write a NOW-NEXT rule that could be used to help determine Darius s wage. d. Write a function rule of the form y = that could be used to calculate Darius s wage after any number of years. e. What will Darius s hourly wage be after 10 years? f. How long will it take Darius to reach the cap of $25 per hour? Show your work or explain your reasoning.
4. If a > b, draw sketches of the graphs of y = a(0.75 x ) and y = b(2 x ). Clearly label your graphs and the values of any x- or y-intercepts. 5. If y = a(0.75 x ) gives the amount of radioactivity y present in a substance after x years have passed, how long will it take for the amount of radioactivity to be half of what it was when x = 0? Show your work or explain your reasoning. 6. Write each of the following expressions as an integer or using radicals in simplest form. a. c. b. 7. Determine whether each of the following statements is true or false. If false, rewrite the right hand side of each statement so that the statement is true. a. (4x) 3 = 12x 3 d. b. e. c.
8. The rule y = 20,000(0.75 x ) gives the value (in dollars) of a car that is x years old. a. Is the value of the car increasing or decreasing? How can you determine this from the rule? b. Does the value of the car change more during the third year or the sixth year? How is your answer reflected in a graph of the function? c. Write a NOW-NEXT rule that can be used to calculate the value of the car after any number of years. d. Write a sentence that describes how the value of the car changes from one year to the next. e. What is the value of the car after 5 years? Show your work or explain your reasoning. f. What question can be answered by solving the equation 2,500 = 20,000(0.75 x )? g. Solve the equation 2,500 = 20,000(0.75 x ) to the nearest tenth of a year. Show your work or explain your reasoning. 9. Consider the following table of values. a. Write a NOW-NEXT rule that matches the pattern of change in the y values. b. Write a rule in the form y = that matches the values in the table.
10. Shown below are graphs of four rules. Without using your calculator, identify the number of the correct graph for each rule in Parts a c. Provide reasoning for each choice. Rule Graph a. y = 30(4 x ) b. y = 10(4 x ) c. y = 30(0.25 x ) 11. Write each of the following in a simpler equivalent form without negative exponents. a. 5x 0 d. b. (3x 2 y) 3 e. 7x 3 c. 12a 4 (2a 8 ) QUADRATIC FUNCTIONS (UNIT 2) 12. Describe how the graphs of y = x 2 + 8x + 5 and y = x 2 + 8x are related to each other. 13. Consider the graph of the equation y = 3x 2 12x. a. Without using your calculator, find the x-intercepts of the graph. b. Without using your calculator, find the minimum or maximum point of this graph.
14. A height of a softball, in feet, that has been pitched by a slow-pitch softball pitching machine is given by the rule h = 16t 2 + 30t + 2.5 for any time t seconds after it is pitched. a. Explain the meaning of the 16, the 30, and the 2.5 in the equation. b. How long is the ball in the air? (Find the time(s) when h=0.) c. What is the maximum height that the ball reaches? When does it reach that height? d. At what time(s) is the ball at least 10 feet above the ground? 15. Chris can buy ice-cream bars for 20 each. Based upon experience she knows that the function rule n = 150 100p will give a good estimate of the number of ice-cream bars she will sell in one day if she charges p dollars for one bar. a. Write a function rule that will give the income Chris can expect if she charges p dollars for each icecream bar. b. Write a function rule that will give the profit that Chris will make each day if she charges p dollars for one ice-cream bar. c. For what price(s) will Chris make at least $15 per day?
d. What is the maximum amount of profit that Chris can make, and what price should she charge to make the maximum profit? e. How many ice-cream bars will Chris sell if she makes the maximum profit? 16. Rewrite each of the following in standard quadratic form. a. x(4x 15) c. (x + 4)(x 3) b. 5x(2x + 1) + 3(2x + 1) d. (x + 6)(x 6) 17. Solve each equation by reasoning with the symbols themselves. a. 3x 2 10 = 17 d. 2x 2 + 5x + 1 = 0 b. 8x 2 6x = 0 e. 6 = x 2 + 3x c. x 2 8x + 12 = 0
FUNCTIONS, SYSTEMS, AND EQUATIONS (UNIT 3) 18. To answer the following, refer to the equation z = where x, y, and z are all positive. a. If x is held constant and y increases, how does z change? What do we call this relationship? b. If y is held constant and x increases, how does z change? What do we call this relationship? c. Write an equivalent rule that shows x as a function of y and z. d. Write an equivalent rule that shows y as a function of x and z. 19. The time required to complete a 100-mile bike race is inversely proportional to the average speed that the rider maintains. a. Write a rule that expresses the relationship between average speed s and race time t. b. What is the constant of proportionality for this situation? c. Tina took 5 hours and 15 minutes to complete the race. What was her average speed? d. Gregory maintained an average speed of 16 miles per hour. How long did it take him to complete the race? 20. Towne Sporting Goods establishes a selling price S for an item based on the cost C that it paid the manufacturer and the rate R of markup that it charges in order to cover its expenses and make a profit. These variables are related by the following equation: S = C(1 + R) a. Towne Sporting Goods gets a pair of in-line skates from the manufacturer at a cost of $80. If Towne uses a 28% markup, what is the selling price of the skates to the nearest dollar?
b. Use the equation S = C(1 + R) to write an equivalent equation that gives C as a function of S and R. c. After the holidays, Towne Sporting Goods had a sale during which it sold all items in the store for a markup of only 10%. The sale price of a tennis racket was $32. To the nearest dollar, how much did it cost Towne Sporting Goods to buy the racket from the manufacturer? 21. Draw a graph of the equation 3x + 4y = 24. 22. Consider the following system of equations: y = 3x 12 6x + 4y = 15 a. Use an algebraic method to solve this system of equations. Show your work. b. How does the solution you found in Part a relate to the graphs of the two equations? 23. Joe looked at the following system of equations and announced that the system had infinitely many solutions. Is Joe correct? Describe how you can determine this just by looking at the equations. 6x + 8y = 24 9x + 12y = 24
24. The Student Council sponsored a talent show to raise money. They charged $5 admission for each adult and $2 for each student. A total of 248 people attended the show and they made $715. How many students and how many adults attended the talent show? 25. A system of linear equations can have 0, 1, or infinitely many solutions. a. Write a system of equations that has no solution. Explain how you know the system does not have a solution. b. Write a system of equations that has exactly one solution. Explain how you know it has exactly one solution. c. Write a system of equations that has an infinite number of solutions. Explain how you know there are an infinite number of solutions. 26. Refer to the following system of linear equations. 3x + 2y = 15 x - y = 2 a. Graph this system of equations and estimate the (x, y) pair that solves it. b. Check your estimate in Part a. Was your estimate correct?
REGRESSION AND CORRELATION (UNIT 4) 27. The scatterplot to the right shows information about newspaper delivery for 13 different people. It shows the number of newspapers on each route n and the amount of time t each person spends delivering the newspapers. The least squares regression line is drawn on the scatterplot and has equation t = 0.6n + 6.3. The correlation for this data is r = 0.65. a. The circled point is an outlier. Describe what type of outlier it is and how this person s newspaper delivery is different from the other s. b. If the circled point was deleted from the data set, how would the slope of the regression line change? c. If the circled point was deleted from the data set, how would the correlation change? Explain your reasoning. d. Add a point to the scatterplot that would be considered an outlier with respect to the number of newspapers only. Give the coordinates of your point. Explain your reasoning. 28. Examine the scatterplots below. The axes all have the same scale. a. Which plot shows the weakest correlation between its two variables? Explain your choice.
b. Do any of the plots illustrate a negative correlation? If so, which plot? Explain your response. c. Which plot shows the strongest correlation? Explain your choice. 29. The scatterplot below gives the number of hours studied last week s and the number of hours of television watched last week t by each student in an eleventh-grade classroom. The equation of the regression line is given below, and the value of the correlation is 0.5462. (Each 2 indicates a single point that represents two students.) t = 8.00 0.341s a. What are the coordinates of the point that appears to be an influential point? If this point was removed from the data set, would the absolute value of the correlation increase, decrease, or remain about the same? b. Briana is the student who studied 12 hours last week and watched 2 hours of television. Graph the regression line on the scatterplot. Draw in the line segment that represents the residual for Briana. Compute the residual for Briana. c. The slope of the regression line is 0.341. Interpret this slope in the context of the data.
d. What might explain the negative correlation between the number of hours these students studied and number of hours they spent watching television? e. The mean number of hours these students spent watching television was 5.28. Describe two ways that you could determine the mean number of hours they spent studying. 30. Match the correct approximate correlation to each plot. I. r = 2.0 II. r = 0.60 III. r = 0.20 IV. r = 0.50 V. r = 0.80 31. Each time Carly filled up the gas tank in her car, she recorded the number of miles she had driven m since the last fill-up and the number of gallons of gas g required to fill her tank. Her data is shown in the scatterplot below. The equation of the linear regression line is g = 0.057m 0.27. a. Describe the shape of this distribution. Include the direction of the relationship, the strength of the relationship and whether the strength varies.
b. What is the approximate slope of the regression line? Interpret the slope in the context of these data. c. Circle the point that could be considered an outlier. Describe the type of outlier that your indicated point is. d. Would the outlier be considered an influential point? Explain your reasoning. e. Which of the following is the correlation for these data? 0.87 0.27 0.057 0.27 0.87 f. Carly was trying to compare her data with that of a friend who lived in Canada. To do this, she changed the miles driven to kilometers driven (1 mile = 1.6 kilometers) and the gallons bought to liters bought (1 gallon = 3.875 liters). Find the correlation for the (kilometers driven, liters used) data. COORDINATE METHODS (UNIT 5) 32. A plan for a new Amusement Park is sketched on a grid below. One unit on the grid is equivalent to 10 meters. The main gate is located at point G. The entrances to some rides are marked: the Ferris wheel is F, the roller coaster is R, and the Tilt-a-Whirl is T. Complete the following tasks about the plan. Show or explain your work for each part.
a. Main Street is planned to run directly from G to F. Find an equation in the form y = ax + b of the line representing Main Street. b. The haunted house H is to be built on Main Street, and it has the same x-coordinate as the roller coaster. Mark H on the map. What are the coordinates of H? c. A concession stand N is planned midway between the gate and the Tilt-a-Whirl T. Mark N on the map, and find its coordinates. d. The planners want the concession stand to be within 100 meters of the roller coaster. Does its present location, found in Part c, satisfy this condition? 33. A parallelogram ABCD has vertex matrix ABCD =. a. Write an equation for the line containing the side of the parallelogram determined by (4, 0) and (6, 3). Show your work. b. Is the parallelogram a rhombus? Justify your response. c. Find equations of the lines containing the diagonals of the parallelogram. Show your work. d. Are the diagonals of the parallelogram perpendicular? Explain your reasoning.
e. Find the midpoint of each diagonal. What do these results tell you about the diagonals? 34. Is the quadrilateral ABCD with vertices A(-3, -2), B(-1, 2), C(4, 3), and D(3, -1) a parallelogram? Provide a mathematical argument that supports your answer. 35. Quadrilateral ABCD is a rhombus. a. Determine the coordinates of point C. Show your work and explain your reasoning. b. Prove that. c. Prove that bisects.
36. Shown below is a circle with radius 4 and center at the origin. Identify the coordinates of two points that are on the circle and are not on the x- or y-axis. Show your work or explain your reasoning. Then, write an equation for the circle. 37. Triangle ABC = is sketched to the right. a. appears to be a right triangle. Check by finding the slopes of and. Is a right triangle? Explain your reasoning. b. Find the lengths of,, and. Explain how to use these lengths to determine whether is a right triangle. c. Write an equation for. Show your work.
d. Draw the image of under each given transformation. Label it A B C. e. Describe the effect on of the transformation represented by. f. Find the area of. Show or explain your work.
g. Consider A B C, the size transformation of magnitude 2 centered at the origin of that you sketched in Part d. Explain how to find the area of A B C from the area of without computing the lengths of any sides of A B C. 38. Consider the transformation that is a composite of these two transformations in the order given. Transformation 1: (x, y) (3x, 3y) Transformation 2: (x, y) (x + 2, y + 1) a. Describe each transformation. b. Suppose that ABC =. What is the image of ABC under this composition of transformations? c. Write a rule (x, y) (, ) that describes this composition of transformations. d. The perimeter of ABC is 12 units. What is the perimeter of the image of ABC? Explain your reasoning or show your work. e. The area of ABC is 6 square units. What is the area of the image of ABC? Explain your reasoning or show your work.
TRIGONOMETRIC METHODS (UNIT 6) 39. Refer to the right triangle ABC. a. If measures 27 and c = 13 cm, find a and b to the nearest hundredth of a centimeter and the measure of to the nearest hundredth of a degree. Show or explain your work. a = b = m b. If a = 12 cm and b = 7 cm, find c to the nearest hundredth of a centimeter and the measures of and to the nearest hundredth of a degree. Show or explain your work. c. c = m m = 40. The Great Pyramid of Cheops in Egypt has a square base 230 meters on each side. Each face of the pyramid makes an angle of approximately 52 with the ground. A sketch of the pyramid is shown below. Show all your work for each problem. a. Find the height of the pyramid. b. Find the area of one triangular face of the pyramid.
41. Suppose four stars in the Milky Way are labeled A, B, C, and D. Star A is 2.1 light years from star B, 3.2 light years from star C, and 1.7 light years from star D. a. Draw a sketch of the stars, labeling each one. b. To the nearest tenth of a light year, how far is star B from star C if m = 71? c. Find the measure of to the nearest degree if star D is 4.2 light years from star C. 42. The pitcher s mound on a softball field is 40 feet from home plate, and the distance between bases is 60 feet. As shown in the diagram to the right, m = 45. (Note: is not a right angle.) a. How far is the pitcher s mound P from first base F? b. Find m.
43. Find the indicated measures for each triangle. a. b. AB = AC = m MK = NONLINEAR FUNCTIONS AND EQUATIONS (UNIT 7) 44. The graph to the right shows the height (in meters) of a baseball in flight as time (in seconds) passes and y = h(x). a. Why is it correct to say that height of a baseball is a function of time in flight? b. Is time in flight of a baseball a function of the height of the baseball? Explain your reasoning. c. What does the equation h(1.2) = 16.8 tell about the flight of the ball? d. What is the value of h(3) and what does it tell about the flight of the ball?
e. Estimate the values of x that satisfy the equation 10 = h(x), and what do those values tell about the flight of the ball? f. Identify the practical domain and range of h(x). g. The maximum value of the graph is 20 and the x-intercepts are (0, 0) and (4, 0). Find a function rule for h(x). h. Identify the theoretical domain and range of h(x). 45. The graph of a particular quadratic function has one of its x-intercepts at (4, 0) and a minimum point of (0, -16). a. What is the other x-intercept of the function? Explain your reasoning. b. Write a function rule for this quadratic function. c. If possible, find a function rule for another quadratic function that has the same x-intercepts as this function but has a different y-intercept. If not possible, explain why not. 46. Write each quadratic expression in equivalent factored form. a. x + 2x 35 c. 2x + 16x + 30 b. x - 8x + 16 d. 3x - 22x + 7
47. Use algebraic reasoning to solve each equation. Show your work. a. x + 7x + 15 = 3 e. log x = 2 b. x - 100 = 0 f. 10 = 83 c. 3x + 5 = - 6x + 9 g. x + 10 = d. 3( ) = 456 48. It is important to understand logarithms as well as be able to use them to solve equations. a. Explain why log 0.1 = -1. b. Explain how you could determine without using a calculator that log 1,863 is between 3 and 4. 49. The number of E. coli bacteria that are present in some food t hours after contamination can be modeled by the function N(t) = 50. a. Identify the theoretical domain and range of N(t).
b. Evaluate N(0) and explain what it tells you about the number of E. coli bacteria present. c. The solutions to 5,000 = 50 answer a question about the number of bacteria present. What is that question? d. Algebraically solve the equation in Part c. Show your work. 50. The tractor-pulling contest is one of the most popular contests each year at the Johnson County Fair. Based on data from previous years, the organizers can expect that income I(p) and expenses E(p) both depend on the price of admission. They predicted that: I(p) = -6 + 100x and E(p) = 15x + 230. a. Use algebraic reasoning to determine the ticket price(s) for which income will equal expenses. b. Write a rule that gives predicted profit F(p) as a function of the admission price. c. Use your profit function to determine the maximum predicted profit. d. What admission price should they charge in order to get the maximum predicted profit?