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1 Copyright 2015 Edmentum All rights reserved. Linear Equations & Graphs 1. A line has a y intercept of and a slope of. Find the equation of the line. A. B. C. D. Evaluate Functions 2. The graph of the function y = f(x) is shown below. What value of x produces f(x) = 3? A. 2 B. 1 C. 1

2 D. 2 Slope 3. Determine the slope of the line above. A. B. C. D. Linear Equations & Graphs

3 Linear Equations & Graphs 4. W. X. Y. Z.

4 A. W B. Y C. Z D. X Evaluate Functions 5. Given the function f(x) = 3x 2 5, find f(h + 2). A. 3h B. 3h + 1 C. 7 D. 3h h + 7 Linear vs Nonlinear 6. y = 9x + 13 Is the equation above linear, nonlinear, both linear and nonlinear, or neither linear nor nonlinear? A. linear B. neither linear nor nonlinear C. nonlinear D. both linear and nonlinear Evaluate Functions 7. Given the function f(x) = x 3 + x 2 7, find f( 4).

5 7. Given the function f(x) = x 3 + x 2 7, find f( 4). A. 55 B. 41 C. 87 D. 73 Linear vs Nonlinear 8. y = 4x Is the equation above linear, nonlinear, both linear and nonlinear, or neither linear nor nonlinear? A. both linear and nonlinear B. nonlinear C. neither linear nor nonlinear D. linear Characteristics of Graphs 9. Alfred graphed two lines, y = x + 6 and y = x 1. Which of the following graphs represents the lines Alfred graphed?

6 W. X. Y. Z. A. Z B. W C. X D. Y Linear Inequalities & Graphs 10. Julian wants to spend no more than $24 at the candy store. The graph below shows different combinations of candy (in ounces) that he can purchase. Which combination could he choose?

7 A. 6 ounces of chocolate and 7 ounces of tarts B. 6 ounces of chocolate and 6 ounces of tarts C. 7 ounces of chocoate and 3 ounces of tarts D. 6 ounces of chocolate and 2 ounces of tarts Solve Linear Equations 11. Which value for x makes the sentence true? 3x 3 = 12 A. 5 B. 4 C. 3 D. 0 Domain & Range 12. Polly is going on a road trip, and she is figuring out what time she will arrive at her destination if she leaves her house at 8:00 in the morning. She needs to determine how long it will take her car to travel fifty miles if she drives 55

8 house at 8:00 in the morning. She needs to determine how long it will take her car to travel fifty miles if she drives 55 miles per hour for the entire trip. In the problem situation above, which variable is the dependent variable? A. Polly is going on a road trip. B. leaving at 8:00 a.m. C. speed of the car D. time it takes to travel fifty miles Relations & Functions 13. Which of the following graphs is not a function? W. X.

9 Y. Z. A. W and X B. W, X, Y and Z C. X, Y, Z D. Y Evaluate Functions 14. Find the value of f( 6) for the function below. A. B. C. D. Evaluate Functions 15. Given the function f(x) = 3x 2 5x, find f(h 2 + 2). A. 3h 2 + 7h + 2 B. 3h 4 + 7h 2 2 C. 3h 4 + 5h D. 3h 4 + 7h 2 + 2

10 D. 3h 4 + 7h Solve Linear Equations 16. What is the value of y when x = 1? A. 1 B. 3 C. 1 D. 2 Slope 17. Which of the statements is true about the two equations below? 1) y = 6x 1 2) y = 6x + 9 A. The equations represent the same line.

11 A. The equations represent the same line. B. The lines are parallel. C. The lines are perpendicular. D. The lines intersect but are not perpendicular. Slope 18. x y What is the slope of the line defined by the table above? A. B. C. D. Linear Inequalities & Graphs 19. Which number line represents the solution to the inequality below?

12 A B C D Evaluate Functions 20. Given the function f(x) = (x + 2) 2 x, find f(n 2). A. n + 2 B. n 2 + n 2 C. n 2 9n + 6 D. n 2 n + 2 Characteristics of Graphs 21. What happens to the graph of y = x when the equation changes to y = x 6? A. The graph shifts down 6 units. B. The graph shifts left 6 units.

13 C. The graph shifts up 6 units. D. The graph shifts right 6 units. Characteristics of Graphs 22. Given the equations f(x) = 3x + 3 and g(x) = 10x + 3, what is the difference between the graphs of f(x) and g(x)? A. The graph of g(x) is 7 units lower than the graph of f(x). B. The graph of f(x) is flatter than the graph of g(x). C. The graph of g(x) is 10 units higher than the graph of f(x). D. The graph of f(x) is steeper than the graph of g(x). Solve Linear Equations 23. Which of the following is a solution to the equation below? 5x = y A. ( 3, 22) B. ( 6, 20) C. ( 10, 19) D. ( 8, 17) Relations & Functions 24. Tara is going to draw a triangle on a coordinate plane. Will this be a function, relation, function and relation, or neither relation nor function? A. function only B. neither relation or function

14 C. relation only D. function and relations Domain & Range 25. The menu from a coffee shop is shown below. Menu Size Price Small $1.93 Medium $2.32 Large $2.80 Extra Large $3.55 What is the independent variable on the menu? A. both size and price B. size C. price D. neither size nor price E. The height of the building is neither an independent nor a dependent variable. Slope 26. What is the slope of the line given by the linear equation above? A.

15 B. C. D. Relations & Functions 27. Does the graph above show a relation, a function, both a relation and a function, or neither a relation nor a function? A. neither a relation nor a function B. both a relation and a function C. relation only D. function only Domain & Range

16 Domain & Range 28. Sally's librarian told her that she could increase the size of her vocabulary by spending more time reading. Based on this information, which statement below best describes the size of Sally's vocabulary? A. The size of Sally's vocabulary is a dependent variable. B. The size of Sally's vocabulary is both an independent and a dependent variable. C. The size of Sally's vocabulary is neither an independent nor dependent variable. D. The size of Sally's vocabulary is a dependent and a constant variable. E. The size of Sally's vocabulary is an independent variable. Linear Inequalities & Graphs 29. Julie has decided to start a snow cone stand, and needs to purchase paper cups and flavored syrup. A local supplier offers packages of 25 cups at a cost of $2.61 per package, and bottles of syrup for $4.06 apiece. She has already decided to purchase 56 bottles of syrup. If her total budget for these supplies is $306.00, and she cannot exceed this budget, which of the following inequalities could be used to solve for p, the number of packages of cups she can purchase? A. $ $2.61p < $ B. $ $2.61p > $ C. $227.36p $2.61 > $ D. $ $4.06p < $ Domain & Range 30. What is the range of the equation y = x 3 11? A. {all real numbers greater than or equal to 11} B. {all real numbers less than or equal to 8} C. {all real numbers less than or equal to 11} D. {all real numbers greater than or equal to 14}

17 D. {all real numbers greater than or equal to 14} Linear Inequalities & Graphs 31. Solve for x. A. B. C. D. Linear Inequalities & Graphs 32. The Marshall High School band is having a car wash to raise money for their trip to the Rose Bowl Parade. The standard donation for a car is $5 and for a truck is $7. The band wants to raise more than $1,400 in standard donations. Which graph shows the number of cars and trucks needed for this goal?

18 W. X. Y. Z. A. X B. Y C. Z D. W Linear Equations & Graphs 33. A.

19 B. C. D. Slope 34. Jolie's cell phone has died, and she needs to use a payphone to make a long distance call. The cost of a longdistance telephone call is shown by the function graphed below. What is the cost per minute to make the phone call? A. $1.15

20 B. $1.20 C. $0.10 D. $0.15 Relations & Functions 35. Which relation diagram represents a function? W. X. Y. Z. A. X B. W

21 B. W C. Z D. Y Solve Linear Equations 36. Which value for x makes the sentence true? 9x 6x = 12 A. 3 B. 4 C. 5 D. 4 / 5 Linear vs Nonlinear 37. X Y , ,776 The table above represents what type of function? A. neither linear nor nonlinear B. nonlinear

22 B. nonlinear C. linear and nonlinear D. linear Solve Linear Equations 38. Solve the following equation for x. 3x 1 = 17 A. 5 B. 10 C. 7 D. 6 Relations & Functions 39. Think about the vertical line test and answer the following question. Would a vertical line be a relation, a function, both a relation and a function, or neither a relation nor a function? A. neither a relation nor a function B. relation only C. both a relation and a function D. function only Linear vs Nonlinear 40. X Y 1 7

23 The table above represents what type of function? A. linear B. nonlinear C. linear and nonlinear D. neither linear nor nonlinear Domain & Range 41. What is the range of the relation above? A. The table does not show a relation. Oklahoma Residential Energy Price Estimates for 2000 (Dollars per Million Btu) Coal $ Natural Gas $7.30 Petroleum Products $11.80 Wood $4.38 Electricity $20.59 Source: U.S. Department of Energy B. {Coal, Natural Gas, Petroleum Products, Wood, Electricity}

24 {Coal, Natural Gas, Petroleum Products, Wood, Electricity} C. {$, $7.30, $11.80, $4.38, $20.59} D. There is no range. Linear Equations & Graphs 42. A. B. C. D. Solve Linear Equations 43. The function f(x) is linear. What is the value of f(3)? x f(x) A. 15 B. 9 C. 19 D. 18

25 Linear vs Nonlinear 44. During the softball game Tonya hit a pop fly that was caught by the centerfielder. Which of the following graphs could represent the ball's path? W. X. Y. Z. A. Graph Z B. Graph W C. Graph Y D. Graph X Linear Equations & Graphs

26 45. Which of the following shows the graph of a line with a slope of 2 and a y intercept of 1? W. X. A. Y B. X Y. Z.

27 C. Z D. W Linear vs Nonlinear 46. y = 6x 2 2 Is the equation above linear, nonlinear, both linear and nonlinear, or neither linear nor nonlinear? A. nonlinear B. both linear and nonlinear C. neither linear nor nonlinear D. linear Slope 47. Determine the slope of the line that passes through the points ( 2, 3) and (5, 1). A. B. C. D.

28 Characteristics of Graphs 48. Find the equation of the line shown above. A. x = 3 B. y = 3 C. x = 3 D. y = 3 Relations & Functions 49. Do the ordered pairs below represent a relation, a function, both a relation and a function, or neither a relation nor a function? A. both a relation and a function B. function only ( 3,4), (0, 2), (5, 12), (8, 18)

29 function only C. relation only D. neither a relation nor a function Characteristics of Graphs 50. What happens to the y intercept of y = x when the function changes to y = x + 4? A. The y intercept changes from 0 to 4. B. The y intercept does not change. C. The y intercept changes to the x intercept. D. The y intercept changes from 0 to 4. Answers 1. D 2. A 3. D 4. A 5. D 6. A 7. A 8. B 9. B 10. D 11. A 12. D 13. D 14. D 15. D 16. D 17. B 18. B

30 18. B 19. D 20. D 21. D 22. B 23. B 24. C 25. B 26. C 27. B 28. A 29. A 30. A 31. B 32. A 33. B 34. D 35. A 36. B 37. B 38. D 39. B 40. B 41. C 42. D 43. A 44. C 45. A 46. A 47. D 48. B 49. A 50. A Explanations 1. Use the given information in the slope intercept form of a line.

31 2. The question is asking what value of x produces f(x) = 3. The x value is an element in the domain. To find an element in the domain, begin by locating the range value, 3, on the y axis. Then move right to where the function touches that grid line. Finally, move down to the x axis to find the domain value. x = 2 3. The slope of a line is represented by the equation below, where the rise is the vertical change in the y values, and the run is the horizontal change in the x values. Choose two points from the line, such as (0, 3), and (5, 1), and find the slope by using the formula.

32 4. 5. Keep in mind the x is just a place holder. So, replace every x in the given function f(x) = 3x 2 5 with h + 2. f(h + 2) = 3(h + 2) 2 5 = 3(h 2 + 4h + 4) 5 = 3h h = 3h h A linear equation is an equation which can be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both 0. The given equation is linear. 7. Keep in mind the x is just a place holder. So, we want to replace every x in the given function f(x) = x 3 + x 2 7 with 4. f( 4) = ( 4) 3 + ( 4) 2 7 = = A linear equation is an equation which can be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both 0. A third degree equation cannot be written in the form Ax + By = C because the exponent on x is greater than 1.

33 A third degree equation cannot be written in the form Ax + By = C because the exponent on x is greater than 1. The given equation is nonlinear. 9. The lines have the same slope, therefore, they are parallel. One has a y intercept of 6 which means it contains the point (0,6). The other has a y intercept of 1 which means it contains the point (0, 1). The only graph depicting two parallel lines with y intercepts of 6 and 1 is W. 10. The total ounces of candy cannot go above the solid line on the graph. There is only one combination that, when plotted, appears in the shaded area x 3 = 12 3x = x = 15 3x 3 = 15 3 x = 5 6 ounces of chocolate and 2 ounces of tarts 12. An independent variable is a variable that can be changed directly and whose change is independent of other variables. A dependent variable is a variable that can only change as a result of changing a different variable. Changes to the dependent variable depend on changes to the independent variable. In this problem situation, the time it takes the car to travel fifty miles depends on the speed of Polly's car. So, time it takes to travel fifty miles is the dependent variable. 13. A relation is a set of one or more ordered pairs. A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. The Vertical Line Test: Given the graph of a relation, if a vertical line can be drawn that does not cross any of the

34 The Vertical Line Test: Given the graph of a relation, if a vertical line can be drawn that does not cross any of the graphs in more than one place, it is a function. Therefore, Y is not a function. 14. Substitute 6 into the given function for. Then, evaluate. 15. Keep in mind the x is just a place holder. So, replace every x in the given function f(x) = 3x 2 5x with h f(h 2 + 2) = 3(h 2 + 2) 2 5(h 2 + 2) = 3(h 4 + 4h 2 + 4) 5h 2 10 = 3h h h 2 10 = 3h 4 + 7h The point on the graph where x = 1 is (1, 2). Therefore, the value of y when x = 1 is Parallel lines have the same slope. The slope of the first and second line is 6. Therefore, the lines are parallel. 18. Slope can be calculated using the following formula.

35 The rise is the change in the y values, and the run is the change in the x values. Choose any two points from the table, and evaluate the formula. Therefore, the slope, as defined in the table, is. 19. Solve the inequality by isolating the variable on one side of the equation. The symbol "<" means less than or equal to. All numbers less than and including 7 will satisfy the equation

36 20. Keep in mind the x is just a place holder. So, replace every x in the given function f(x) = (x + 2) 2 x with n 2. f(n 2) = ((n 2) + 2) 2 (n 2) = n 2 n When a number is added or subtracted inside of the absolute value sign, it causes a horizontal shift to the absolute value function. If the number is added, the graph shifts left, and if the number is subtracted, the graph shifts right. Since 6 is subtracted from the equation, the graph shifts right 6 units. 22. The slope intercept form for the equation of the line is y = mx + b, where m is the slope of the line, and b is the y intercept of the line. Compare the slope and the y intercept of the two given equations. The y intercept, b = 3, is the same in both the graphs of f(x) = 3x + 3 and g(x) = 10x + 3. So, the only difference is that the slope, m = 3, of the graph of f(x) is smaller than the slope, m = 10, of the graph of g(x). Thus, the difference between the two graphs of f(x) = 3x + 3 and g(x) = 10x + 3, is that the graph of f(x) is flatter than the graph of g(x). 23. For a coordinate (x,y) to be a solution, it must hold true when the values are substituted into the equation. First, try ( 10, 19). 5( 10) = ( 19) 50 = So, ( 10, 19) is not a solution. Next, try ( 3, 22).

37 Next, try ( 3, 22). 5( 3) = ( 22) 15 = So, ( 3, 22) is not a solution. Now, try ( 8, 17). 5( 8) = ( 17) 40 = So, ( 8, 17) is not a solution. Finally, try ( 6, 20). 5( 6) = ( 20) 30 = = 30 Therefore, ( 6, 20) is a solution. 24. A relation is a set of one or more ordered pairs. A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. No matter which way the triangle is turned, there will be more than one domain with two ranges. The shape will not pass the vertical line test; therefore, it is a relation only. 25. An independent variable is a variable whose value determines the value of other variables, and it is changed in an experiment to see its effect on other variables. A variable that changes because of a change in the independent variable is a dependent variable.

38 Looking at the menu, changing the size of the drink ordered changes the price of the drink. Therefore, the independent variable on the menu is size. 26. The given equation of the line is currently written in the standard form. In order to find the slope of the line, rewrite the equation in the slope intercept form, the slope of the line and b is the y intercept. The slope will be the coefficient of x., where m is Therefore, the slope is. 27. A relation is a set of one or more ordered pairs. A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. In this case, there is one y coordinate for every x coordinate. The vertical line test can be used to determine this. Therefore, it is both a relation and a function. 28. An independent variable is a variable whose value determines the value of other variables, and it is changed in an experiment to see its effect on other variables. A variable that changes because of a change in the independent variable is a dependent variable. According to the librarian, changing the amount of time Sally spends reading will change the size of her vocabulary. Therefore, the size of Sally's vocabulary is a dependent variable. 29. First, determine the total cost of the syrup Julie intends to purchase. The cost of the syrup plus the cost of the cups must be less than or equal to her budget of $

39 ($ ) + ($2.61 p) < $ First, find the value of x that will make x 3 = 0, x = 3. $ $2.61p < $ Knowing x 3 = 0 when x = 3, find y when x = 3. When x = 3, y = 11, showing that either a minimum or a maximum occurs at (3, 11). The given function is a positive absolute value function, meaning that the graph opens upward and that (3, 11) is a minimum. Therefore, the range of the function is {all real numbers greater than or equal to 11}. 31. Keep in mind that the goal is to isolate x. 32. In each graph, the number of cars washed ($5 each) is represented by x and the number of trucks washed ($7 each) is represented by y. Set up the situation as an inequality as shown below and solve for y.

40 The correct graph should have a y intercept of 200 and a slope of. The correct graph should also shade above the dotted line since the greater than (>) symbol was used. The only graph that meets these three conditions is X. 33. To find the equation, first find the slope of the line. For any two points (x 1, y 1 ) and (x 2, y 2 ), the slope is given by the following. Now, use one of the given points, the slope, and the point slope form of an equation of a line. 34. The cost per minute on a graph is found by looking at how much the cost is constantly increasing from one minute to the next. Since the phone call costs $1.35 after 1 minute, $1.50 after 2 minutes, and $1.65 after 3 minutes, the phone call is increasing in cost by $0.15 every minute.

41 Thus, the cost per minute for the phone call is $ For a relation to be a function, each input value can only correspond to one output value. The relation diagram where each input value has exactly one arrow drawn to an output value will represent a function. Therefore, diagram X represents a function x 6x = 12 3x = 12 3x 3 = 12 3 x = In a nonlinear function, numbers do not increase at a constant rate. In this problem, the numbers in the Y column are 6, 6 2, 6 3, 6 4, and 6 5. So, the numbers are given by the exponential equation y = 6 x. The function is nonlinear. 38. Keep in mind that the goal is to get x by itself. 39. A relation is a set of one or more ordered pairs. 3x 1 = 17 3x = x = 18 3x 3 = 18 3 A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. The Vertical Line Test: Given the graph of a relation, if a vertical line can be drawn that does not cross any of the graphs in more than one place, it is a function. If the relation being tested is a vertical line, then any x in the domain of the relation (which there would be only x = 6

42 If the relation being tested is a vertical line, then any x in the domain of the relation (which there would be only one) will correspond with every y of the range (an infinite number of points). So, a vertical line can be drawn that crosses the graph in more than one place (the vertical line itself). Therefore, a vertical line is not a function, and it is a relation only. 40. In a nonlinear function, numbers do not increase at a constant rate. In this problem, the numbers in the Y column are 2(1) 2 + 5, 2(2) 2 + 5, 2(3) 2 + 5, 2(4) 2 + 5, and 2(5) So, the numbers are given by the quadratic equation y = 2x The function is nonlinear. 41. The table shows a relationship between an energy source and the price per million Btu. In this case, for every energy source (element of the domain), there is a corresponding price (element of the range). Therefore, the range is {$, $7.30, $11.80, $4.38, $20.59}. 42. Use the point slope form of an equation of a line to find the equation. Substitute the point and slope into the equation. 43. In a linear function, the output values have a constant difference. In this function, the output values have a difference of 4. Also note, as the x values increase, the values of f(x) increase. When x equals 3, the corresponding value of f(x) will be 4 more than the value of f(x) when x equals 2.

43 value of f(x) when x equals 2. Therefore, when x equals 3, the value of f(x) equals The ball would not accelerate (or decelerate) at a constant rate, therefore its path cannot be represented by a straight line. Because the ball not only flies into the air, but also returns to the ground, its flight path is represented by a parabola. Graph Y is the only nonlinear function showing both the ball's ascent and descent. 45. Given that the y intercept is 1, the line passes through the point (0,1). This means that either X or Y is the correct answer. Since slope = left. and the slope given in the question is 2, the line will rise 2 units for every 1 unit it moves to the Of the remaining possible correct answers, only graph Y has a slope of 2. Therefore, the correct answer is Y. 46. A linear equation is an equation which can be written in the form Ax + By = C where A, B, and C are real numbers and A and B are not both 0. A second degree equation cannot be written in the form Ax + By = C because the exponent on x is greater than 1. The given equation is nonlinear. 47. Given any two points (x 1, y 1 ) and (x 2, y 2 ), the slope of the line passing through the points is given by the following formula. 48. Since the line is a horizontal line, the equation of the line will be in the form y = k. Since the line passes through the point (0, 3), the equation of the line is y = A relation is a set of one or more ordered pairs.

44 A function is a relation in which each element of the domain is paired with EXACTLY one element of the range. In this case, there is one y coordinate for every x coordinate. The vertical line test can be used to determine this. Therefore, it is both a relation and a function. 50. The equation for a line can be written as y = mx + b, where b is the y intercept. When y = x, b = 0. When y = x + 4, b = 4. Therefore, the y intercept changes from 0 to 4.