MATH 021 TEST 2 REVIEW SHEET

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1 TO THE STUDENT: MATH 021 TEST 2 REVIEW SHEET This Review Sheet gives an outline of the topics covered on Test 2 as well as practice problems. Answers for all problems begin on page 8. In several instances, it is suggested that you go over definitions used in this course. Definitions can be found in the Glossary/Index listed on pages A-49 to A-56 of your textbook. I. Equations 1. What is a proportion? Explain how to solve a proportion. 2. How can you tell if a proportion is a true statement? 3. Solve each equation, and check your solution. a. f. g. c. h. i. e. j. II. Inequalities 4. Write the following inequality symbolically; then graph it on a number line. x is less than Write the following inequality symbolically; then write it in words. 6. Write the following inequality in words; then graph it on a number line. 1

2 7. When solving an inequality, when is it necessary to reverse the direction of the inequality? 8. Solve each inequality algebraically. Then graph the solution on a number line. a. c. III. Formulas 9. Solve each formula for variable indicate a. c. IV. Functions 10. State the definition of a function. 11. Fill in the Blank: The Vertical Line Test states that if a line crosses a graph in more than one place, the graph is not a function. This is because for some value of, there would be more than one value for. 12. State whether each of the following represent a function? Explain why. a. x y 2

3 c Suppose you are renting a storefront for your business. The input is the size of the store, s, in square feet. The output is the monthly rent, R, so we write R= f(s). a. Interpret in the context of this situation. If the monthly rent for 1,500 square feet is $3000, write this fact using function notation. 14. The amount of money in dollars, m, you make at a job depends on the number of hours, h, you work. V. Slope a. In this situation, the input is. The output is. c. We say is a function of. Write the relationship using function notation. 15. What does slope tell you about the graph of a line? 16. Write the formula for the slope of a line given two points. 17. Does a horizontal line have a slope? Explain. 18. Does a vertical line have a slope? Explain. 19. Explain how to tell from a graph if a line has positive slope. 20. Explain how to tell from a graph if a line has negative slope. 21. Find the slope of the line through each pair of points. Then determine whether the line is rising, falling, horizontal, or vertical. a. c. 3

4 22. The following table gives the cost of an international telephone call. Assume there is a linear relationship. Find the slope, and include the correct units. Input: minutes Output: cost in dollars 30 $ $9.15 VI. Intercepts 23. What is an x-intercept? How do you find an x-intercept from an equation? 24. What is a y-intercept? How do you find a y-intercept from an equation? 25. Find the x- and y-intercepts of each equation. a. 26. On graph paper, use the x- and y-intercepts to graph each equation from problem # Find the x- and y-intercepts from the graph. 28. For the function. a. Find. Write your answer as an ordered pair. Solve. Write your answer as an ordered pair. c. Find. Write your answer as an ordered pair. What is the name of this special point? 4

5 Find x if. Write your answer as an ordered pair. What is the name of this special point? 29. The function gives the profit from sales of music CDs for a small music company, where x is the number of CDs sol a. Find. Give the practical meaning in the context of the situation. Solve. Give the practical meaning in the context of the situation. VII. Equations of Lines 30. What is the slope-intercept formula for the equation of a line? What do the m and b represent? 31. Write each equation in slope-intercept form. Then identify the slope and y-intercept. a. 32. On graph paper, use the slope and y-intercept to graph each equation from problem # Find the equation of the line having the given slope and point. a. 34. Find the equation of the line through the two points given. a. c. 35. Find the slope, y-intercept, and equation for each line. a. 5

6 36. a. On graph paper, graph the line with slope that passes through (8, 2). Find the y-intercept. c. Label the y-intercept and at least two other points on the line. Write the equation of the line. 37. Write the equation of the horizontal line that passes through (3, 1). 38. Write the equation of the vertical line that passes through ( 2, 1). 39. Fill in the Blank: If two lines are parallel, their slopes must be. 40. True or False: The lines and are parallel. (Hint: Write each line in slope-intercept form before deciding.) 41. A health club requires a deposit of $520, and then $10 is deducted for each workout. a. Define input and output variables that can be used to find the amount remaining on the $520 deposit. Write the equation that represents the amount remaining on the deposit. c. What is the slope? What is the practical meaning of the slope in the context of the situation? What is the y-intercept? What is the practical meaning of the y-intercept in the context of the situation? e. What is the x-intercept? What is the practical meaning of the x-intercept in the context of the situation? f. On graph paper, graph this relationship. Be sure to include an appropriate scale and to label your axes. 42. The Ajax Car Rental cost schedule for a compact car is shown on the table. The cost is also shown in the graph. Miles: x Cost: y 50 $ a. What is the input or independent variable? What is the output or dependent variable? 6

How many miles may be driven for $45? 43. To produce 50 copies of the school newspaper, the cost per paper is $0.26. To produce 200 copies of the school newspaper, the cost per paper is $0.20. Find the linear equation that fits these data.

7 c. What is the slope of the line? What is the practical meaning of the slope in the context of this situation? What is the y-intercept? What is the practical meaning of the y-intercept in the context of this situation? e. What is the equation of the line? f. What is the cost of driving 175 miles? g. How many miles may be driven for $45? 43. To produce 50 copies of the school newspaper, the cost per paper is $0.26. To produce 200 copies of the school newspaper, the cost per paper is $0.20. Find the linear equation that fits these data. (Assume that the cost is a function of the number of copies.) 44. Match the equation (a) (d) with the correct graph (A) (D). a. c. VIII. Review from Unit 1 (Note: In addition to the following problems, you should also review your notes and other materials from Unit 1.) 45. Combine like terms: 46. Evaluate. 7

8 47. Use the table to solve each equation. x a. c. What can you say about the solution to? 48. Use the graph to solve each equation. Note that equations may have more than one solution or no solution. a. c. 49. a. Build an input-output table for the equation. Use integer inputs from 1 to 3. On graph paper, graph the ordered pairs from your table and connect them to form a smooth curve. 50. Write each sentence as an equation with x as the input and y as the output. a. The output is ten more than triple the input. The output is two less than the quotient of the input and 9. 8

9 ANSWER KEY 1. A proportion is an equation formed by two equal ratios. To solve a proportion, find both crossproducts and set them equal. Then solve the resulting equation. 2. A proportion is a true statement if the result of cross-multiplication is the same number. 3. a. c. 4. e. f. g. h. i. j. 8. c. 9. a. c. or 10. A function is a relationship where for each input there is exactly one output. 11. vertical, x, y 5. x is greater than or equal to x is between -2 and 4, including Reverse the direction of the inequality when you multiply or divide both sides of the inequality by a negative number. 8. a. 12. a. Yes, this is a function. The graph passes the Vertical Line Test. For each input, there is exactly one output. No, this is not a function. The graph does not pass the Vertical Line Test. For some input, there is more than one output. c. Yes, this is a function. For each input, there is exactly one output. No, this is not a function. The input 1 has more than one output. 13. a. For a 2000 square foot store, the monthly rent is $ a. the number of hours worked the amount of money earned c. the amount of money earned, the number of hours worked 9

10 15. The slope tells you the steepness and direction of the line. 26. a Yes, a horizontal line has a slope. The slope of any horizontal line is zero. This is because the difference in y-values (the numerator in the equation for #16) is zero. 18. No, a vertical line does not have a slope. We say the slope of any vertical line is undefined, or we say there is no slope. This is because the difference in x-values (the denominator in the equation for #16) is zero. Recall that division by zero is undefine 19. The slope is positive if the line is rising (from left to right). 20. The slope is negative if the line is falling (from left to right). 21. a., rising, horizontal c. no slope, vertical, falling 22. m = $0.07/min 23. An x-intercept is a point where the graph crosses the x-axis. To find the x-intercept from an equation, you let and solve for x. 24. An y-intercept is a point where the graph crosses the y-axis. To find the y-intercept from an equation, you let and solve for y. 25. a. x-intercept: y-intercept: x-intercept: y-intercept: 27. x-intercept: y-intercept: 28. a. ; ; c. ; ; y-intercept ; ; x-intercept 29. a. If the company does not sell any CDs, then it will have a loss of $ If the company sells about 158 CDs, then it will not have a profit or loss. The company will break even. 30. ; m is the slope; b is the y-intercept. 31. a. ; ; ; ; 10

11 32. a. 36. a. c. See graph equal 40. False. The slope of the first line is 3, but the slope of the second line is a. 34. a. 34. c. 41. a. Let x = the number of workouts. Let y = the amount remaining. or c.. $10 is deducted for each workout. or. The initial deposit is $520. e. or. After 52 workouts, you have no money remaining. 35. a. ; ; ; ; 11

12 41. f. 47. c. x is between -12 and a. c. No solution 49. a. 42. a. the number of miles the cost c. m = $0.08/mi. For each mile driven, there is an additional cost of $0.08. or. There is an initial cost of $35. e. f. $49 g. 125 miles 43. Let x = the number of copies Let y = the cost per copy Then: x y a. C A c. D B a. 50. a. 12

Section 1.3. Slope formula: If the coordinates of two points on the line are known then we can use the slope formula to find the slope of the line.

Section 1.3. Slope formula: If the coordinates of two points on the line are known then we can use the slope formula to find the slope of the line. MATH 11009: Linear Functions Section 1.3 Linear Function: A linear function is a function that can be written in the form f(x) = ax + b or y = ax + b where a and b are constants. The graph of a linear