Integrated I Final Exam Review

Transcription

1 Name: Integrated I 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. Remember to use your toolkit entries to help you if you need it!! 1. Match each description with an appropriate graph(s). Write a sentence that explains the reasoning behind each choice. i. y = ax + b if both a and b are greater than zero. ii. y = ax + b if a is less than zero and b is greater than zero. 2. Michelle participated in a walk for charity. Her mother donated $5 and then gave an additional $2 for each mile that Michelle completed during the 3-hour walk. a. Complete the table below to indicate how much Michelle s mother owed as a function of the distance Michelle walked. b. Write a rule relating NOW and NEXT that shows how the amount Michelle s mother owed changed with each additional mile that Michelle walked. NEXT =, starting at

2 3. David and Isabella are planning a family reunion. They are trying to decide between two locations. Old Mill Pond will charge them $300 for the use of the grounds and $6.25 per person for refreshments. Long Lake Park will charge them $200 for use of the grounds but will charge them $7.50 per person for refreshments. a. For what number of people will the two locations cost the same? Explain your reasoning or show your work. b. If they are expecting 125 people to attend the party, which location will cost less? Explain your reasoning. 4. Solve each inequality. a. 6x 24 > 8x b. 28 4x < Rewrite each of the following in the shortest form possible. a. 7x x d. 19 8x + 3(x + 7) b. 2(5x + 8) e. 5x 3(4 + 2x) + 10 c. 3 (6x 5)

3 6. The scatterplot below shows the relationship between the number of students attending school and the number of teachers in ten states of the United States. (Source: The World Almanac and Book of Facts, 2006.) a. On the scatterplot to the right, draw a line that you believe is a good fit for this data. b. Write a rule that represents the line you drew in Part a. Use S for the number of students and T for the number of teachers. Show or explain you work. Rule: c. What is the slope of your line? Explain the meaning of the slope of your line in this context. Slope of line: d. Use your line to predict the number of teachers in a state that has 400,000 students. Explain how you made your prediction. Number of Teachers: e. Use your rule to predict the number of teachers in a state with 400,000 students. Show or explain your work. Number of Students:

4 7. The graph below indicates the value of a small used car as a function of how old the car is. a. Estimate the slope and y-intercept of the line. Explain how you made each estimate. Estimate of the slope: Estimate of the y-intercept: b. Write a function rule that would match the graph. Function rule: c. Explain the meaning of the slope and the y-intercept in terms of the value of the car and the age of the car. Slope: y-intercept: 8. Nicole measured the temperature of water in a beaker as it heated. After drawing a scatterplot of her data, Nicole drew a modeling line that fit the data well. Her line showed how the temperature of the water changed over time. The function rule for her line was T = s, where T is the temperature in C after s seconds. a. Draw Nicole s modeling line on the grid to the right. Explain the method you used to draw the line with equation T = s.

5 b. What does the 15 in Nicole s rule tell you about the temperature of the water? c. What does the 0.5 in Nicole s rule tell you about the temperature of the water? d. Use Nicole s rule to estimate the temperature of the water after 120 seconds. Explain your reasoning or show your work. e. Water boils at a temperature of 100 C. How long did it take for the water to boil? Explain your reasoning or show your work. 9. Use algebraic reasoning to solve each of the following equations. You may use your calculator for arithmetic. Show your work or explain your reasoning and check your solution. a. 100 = 10x + 5 c. 3(x + 5) = 60 b x = 260 d. 5x 12 = 2x Write a rule in the form y = for the line that passes through ( 2, 13) and (4, 5). Show your work or explain your reasoning.

6 11. Suppose that x 3 = 64 and x 5 = 1,024 and (64)(1,024) = x n. Find the value of n. Explain your reasoning. 12. Are 3(4 x ) and 12 x equivalent for all x? Explain your reasoning. 13. 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.

7 14. 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. 15. Write each of the following expressions as an integer or using radicals in simplest form. a. c. b. 16. 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 17. 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. 18. 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.

9 19. 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 ) 20. 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 ) 21. Describe how the graphs of y = x 2 + 8x + 5 and y = x 2 + 8x are related to each other.

10 22. Consider the graph of the equation y = 3x 2 12x. a. Without using your calculator, find the x-intercepts of the graph. Show your work. b. Without using your calculator, find the minimum or maximum point of this graph. Show your work. 23. 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 t 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? (Meaning, what will the time be when the height is 0?) Show you work. c. What is the maximum height that the ball reaches? When does it reach that height? Show your work. 24. Chris can buy ice-cream bars for 20 each. Based upon experience she knows that the function rule n = p 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 ice-cream bar.

11 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? 25. 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) 26. Solve each equation by reasoning with the symbols themselves. a. 3x 2 10 = 17 c. 8x 2 6x = 0

12 c. x 2 8x + 12 = 0 e. 6 = x 2 + 3x d. 2x 2 + 5x + 1 = Below is a histogram of the estimated population (in thousands) of the 50 largest cities in the United States for the year (Source: The World Almanac and Book of Facts World Almanac Books, New York, NY, 2006) a. How many of these cities had populations of less than 1,000,000? b. Estimate the median of the data set. Explain your process for estimating the median. c. Would you expect the median to be larger, the same, or smaller than the mean? Explain your response.

13 d. Describe the distribution (shape, center, and spread) of the population of the 50 largest cities in the United States in e. New York City had a population of 8,104,798 in Suppose the population of New York was deleted from this data set and a new mean and median were computed. How would the new mean and median compare to the original mean and median? Explain your reasoning. 28. A set of data with 50 data values is displayed in the histogram below. a. Indicate the order, from smallest to largest, of the mean, median, and mode of this distribution. Explain your reasoning. b. Could the box plot below be a box plot of the same data that is displayed in the histogram in Part a? Explain your reasoning. 29. The manager of the concession stand at the school football games keeps track of the total food and drink sales during each game of the season. The mean food and drink sales per game for the last year was $550 and the standard deviation was $75. The manager hopes to increase sales in the coming year. a. If the manager is able to increase sales each game by $50, what will the new mean and standard deviation be? b. If the manager is able to increase sales each game by 10%, what will the new mean and standard deviation be?

14 30. In the Hamilton family, the mean age is 32 years with a standard deviation of 23.4 years. a. If you convert each of their ages to months, what will be the mean and standard deviation? b. In 5 years, what will be the mean and standard deviation of their ages, in years? (Assume no births or deaths have occurred.) 31. The histogram below shows cholesterol measurements for a group of people being treated for high cholesterol. a. Describe the shape of this distribution. b. Estimate the mean. Explain your reasoning. c. Estimate the standard deviation. Explain your reasoning. d. The five-number summary of these measurements is 207, 241.5, 255, 267.5, 311. Show mathematically whether the maximum is or is not an outlier.

15 32. Decide if each of the following statements is True or False. In either case, provide reasoning to support your answer. a. It is possible to draw a triangle with side lengths of 8 in., 15 in., and 7 in. True False b. A triangle with sides of length 5 cm, 13 cm, and 12 cm is a right triangle. True False c. Every triangle can be used to tile the plane. True False d. Every right pyramid with a regular octagonal base has exactly one axis of rotational symmetry. True False e. Every right prism with a regular hexagon as a base has exactly six planes of symmetry. True False

16 33. a. Sketch a kite that has two sides of length 5 cm and two sides of length 7 cm. Clearly label the side lengths. b. Keisha also drew a kite using two sides of length 5 cm and two sides of length 7 cm. Will Keisha s kite be congruent to yours? Explain your reasoning as carefully as possible. 34. The figure below is a parallelogram with diagonal. a. Use the definition of parallelogram to explain why ABD CDB. b. Is a symmetry line for the parallelogram? Why or why not? c. Identify any rotational symmetry for parallelogram ABCD. If it does not have any rotational symmetry, explain why not. d. If AB = 5 and BD = 13, then AD < and AD >. e. If AB = 5 and BD = 13, what would the length of need to be in order for ABCD to be a rectangle? Explain how you know it would be a rectangle. 35. a. Draw a three-dimensional figure that has plane symmetry.

17 b. Describe or draw one symmetry plane for your figure. c. Does your figure have any rotational symmetry? If so, identify one axis of rotation. If not, explain why not. 36. ABCD is a rectangle and M is the midpoint of. a. Is ABM DCM? Explain your reasoning. b. If CD = 6 cm and CM = 10 cm, find the area of rectangle ABCD. 37. Line l is a symmetry line for quadrilateral WXYZ. a. Identify any rotational symmetry for WXYZ. Explain your reasoning. b. If m X = 125, find the measure of the remaining three angles of quadrilateral WXYZ. Explain your reasoning. m W = m Y = m Z = Reasoning:

18 c. Quadrilateral WXYZ is reproduced below. Add line segments to the drawing below so that is a symmetry line for a new figure. d. Find the sum of the interior angles for the completed figure in Part c. Show your work or explain your reasoning. e. Find the sum of the exterior angles for the completed figure in Part c. Show your work or explain your reasoning. 38. The Quintana family bought a grandfather clock to put in the entryway. It came in a rectangular box with dimensions of 1.5 feet by 2 feet by 6 feet. a. Make a sketch of the three-dimensional box. Label all of the lengths of the edges. Find the length of the longest diagonal of a face of the box. b. Sketch or describe all symmetry planes of the box. Explain how these are related to the symmetries of its faces. c. Sketch a net that could be used as a pattern for the manufacture of the box.