Which Table Represents a Linear Function

4-7 Linear and Nonlinear Functions Determine whether each table represents a linear or nonlinear function. Explain. 1. SOLUTION: Determine how the values change for x and y. As x increases by 2 each time, y increases by 1 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 2, y increases by 1. 2. SOLUTION: Determine how the values change for x and y. Although x increases by 1 each time, y increases by a larger amount each time. The rate of change is not constant, so this function is nonlinear. ANSWER: Nonlinear; rate of change is not constant. Page 1 4-7 Linear and Nonlinear Functions 3. SOLUTION: Determine how the values change for x and y. As x increases by 5 each time, y increases by 15 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 5, y increases by 15. 4. SOLUTION: Determine how the values change for x and y. Although x increases by 2 each time, y decreases by a larger amount each time. The rate of change is not constant, so this function is nonlinear. ANSWER: Nonlinear; rate of change is not constant. Page 2 4-7 Linear and Nonlinear Functions 5. The Guzman family drove from Anderson to Myrtle Beach . Use the table to determine whether the distance driven is a linear function of the hours traveled. Explain. SOLUTION: Determine how the values change for x and y. The rate of change is constant. As the time increases by 1 hour, the distance increases by 65 miles, so this function is linear ANSWER: Yes; the rate of change is constant; as the time increases by 1 hour, the distance increases by 65 miles. Page 3 4-7 Linear and Nonlinear Functions 6. The table shows the height of several buildings in Chicago. Use the table to determine whether the height of the building is a linear function of the number of stories. Explain. SOLUTION: Determine how the values change for x and y. Although x increases by 5 each time, y changes by different amounts each time. The rate of change is not constant, so this function is nonlinear. ANSWER: No; the rate of change is not constant. Page 4 4-7 Linear and Nonlinear Functions 7. There are 3,600 seconds in one hour. The total seconds is a function of the hours. Does this situation represent a linear or nonlinear function? Explain. SOLUTION: Make a table with two rows and five columns. Label the first row Hours(h) and the second row Seconds(s). Choose values for h and evaluate them in the expression z = 3,600h. Hours (h) Seconds (s) 1 2 3 4 3,600 7,200 10,800 14,400 Graph the points (h, s) on a coordinate plane. The ordered pairs (hours, seconds) lie on a straight line, so the situation represents a linear function. ANSWER: Linear; sample answer: If you graph the function, the ordered pairs (hours, seconds) lie on a straight line. Page 5 4-7 Linear and Nonlinear Functions 8. A football is placed on the ground to kick a field goal. The height of the ball is a function of the time in seconds. Does the path the football follows after being kicked represent a linear or nonlinear function? Explain. SOLUTION: Nonlinear; sample answer: After being kicked, the ball will reach a maximum height and come back to the ground. ANSWER: Nonlinear; sample answer: After being kicked, the ball will reach a maximum height and come back to the ground. Graph the function by making a table of ordered pairs. Determine whether the function is linear or nonlinear. Explain. 9. y = –x + 1 SOLUTION: Select any four values for the domain x. Substitute these values for x to find the value of y. Graph each ordered pair. Draw a line that passes through each point. Sample Table: x –x + 1 y (x, y) –2 –(–2) + 1 3 (–2, 3) 0 –(0) + 1 1 (0, 1) 1 –(1) + 1 0 (1, 0) 2 –(2) + 1 –1 (2, –1) Graph: Page 6 4-7 Linear and Nonlinear Functions Linear; sample answer: The points lie on a straight line. ANSWER: Linear; sample answer: The points lie on a straight line. 10. y = SOLUTION: Select any four values for the domain, x. Substitute these values for x to find the value of y. Graph each ordered pair. Draw a line that passes through each point. Sample Table: x y (x, y) –3 Page 7 4-7 Linear and Nonlinear Functions 0 Undefined Undefined 3 6 Graph: Nonlinear; sample answer: The graph is a curve. ANSWER: Nonlinear; sample answer: The graph is a curve. 11. y = Page 8 4-7 Linear and Nonlinear Functions SOLUTION: Select any four values for the domain, x. Substitute these values for x to find the value of y. Graph each ordered pair. Draw a line that passes through each point. Sample Table: x y (x, y) –4 –6 (–4, –6) –2 –3 (–2, –3) 0 0 (0, 0) 4 6 (4, 6) Graph: Linear; sample answer: The points lie on a straight line. ANSWER: Linear; sample answer: The points lie on a straight line. 12. Identify Structure Complete the graphic organizer by determining if the graphs represent linear or nonlinear functions. Page 9 4-7 Linear and Nonlinear Functions SOLUTION: Linear functions have graphs that are straight lines. The first graph is the only graph that is a straight line. The rest are nonlinear. ANSWER: Page 10 4-7 Linear and Nonlinear Functions 13. Persevere with Problems Does the graph shown represent a linear function? Explain. SOLUTION: The graph is a line, however, the graphs of vertical lines are not functions because there is more than one value of y that corresponds to x = 2. So, the graph does not represent a linear function. ANSWER: No; sample answer: the graphs of vertical lines are not functions because there is more than one value of y that corresponds to x = 2. 14. Model with Mathematics Give an example of a situation that can be represented by a nonlinear function. SOLUTION: Sample answer: Every hour the number of bacteria in a petri dish doubles. ANSWER: Sample answer: Every hour the number of bacteria in a petri dish doubles. Page 11 4-7 Linear and Nonlinear Functions 15. Reason Inductively Explain how you can use different representations to determine whether a function is linear. SOLUTION: Sample answer: A non-vertical graph that is a straight line is linear. An equation that can be written in the form y = mx + b is linear. If a table of values shows a constant rate of change, it is linear. ANSWER: Sample answer: A non-vertical graph that is a straight line is linear. An equation that can be written in the form y = mx + b is linear. If a table of values shows a constant rate of change, it is linear. Determine whether each table represents a linear or nonlinear function. Explain. 16. SOLUTION: Determine how the values change for x and y. As x increases by 2, y increases by a greater amount each time. The rate of change is not constant, so this function is nonlinear. ANSWER: Nonlinear; rate of change is not constant. Page 12 4-7 Linear and Nonlinear Functions 17. SOLUTION: Determine how the values change for x and y. As x increases by 4 each time, y decreases by 3 each time. The rate of change is constant, so this function is linear. ANSWER: Linear; rate of change is constant; as x increases by 4, y decreases by 3. 18. Copy and Solve The area of a square is a function of its perimeter. Graph the function on a separate sheet of grid paper. Explain whether the function is linear and if the graph is increasing or decreasing. SOLUTION: Create a table of values that has two rows and five columns. Label the rows for perimeter P and area A. Choose a side length for each square and then calculate the perimeter and area to complete the table. Use the formulas P= 4s and A = s2, where P is the perimeter of a square, A is the area of a square, and s is the side length of the square. Perimeter P 0 4 8 16 0 1 4 16 (units) Area A (sq. units) Graph the points as (P, A) on a coordinate plane. Page 13 4-7 Linear and Nonlinear Functions Sample answer: The function is nonlinear because the graph of the function is not a straight line and the line is increasing.. ANSWER: Sample answer: The function is nonlinear because the graph of the function is not a straight line and the line is increasing. 19. Multiple Representations Recall that the circumference of a circle is equal to two times π times its radius and that the area of a circle is equal to π times the square of the radius. a. Tables Complete the table showing the circumference and area of circles with radius r. Page 14 4-7 Linear and Nonlinear Functions b. Graphs Graph the ordered pairs (radius, circumference) and (radius, area) for each function on the same coordinate plane. c. Words Is the circumference of a circle a linear or nonlinear function of its radius? the area? Explain your reasoning. SOLUTION: a. Substitute the values for the different radii into the expressions and simplify. Page 15 4-7 Linear and Nonlinear Functions b. Labe the x-axis as radius. Label the y-axis as circumference/area. Graph the ordered pairs (radius, circumference) then connect those points. Then graph the ordered pairs (radius, area) and connect those points. c. When you connect the points for the circumference, the graph is straight line so the function is linear. When you connect the points for area, they form a curve so the function is nonlinear. ANSWER: a. b. Page 16 4-7 Linear and Nonlinear Functions c. Circumference: linear; sample answer: When the ordered pairs are graphed, the points fall in a line. Area: nonlinear; sample answer: When the ordered pairs are graphed, the points do not fall in a line. 20. Determine if each table represents a linear or nonlinear function. SOLUTION: This table represents a linear function because the x-values increase by 2.5 each time and the y-values increase by 0.75 each time. Page 17 4-7 Linear and Nonlinear Functions This table represents a linear function because the x-values decrease by 4 each time and the y-values increase by 5 each time. This table represents a nonlinear function because the x-values increase at a constant rate of 3, but the y-values increase by a greater amount each time. This table represents a linear function because the x-values increase by 2 each time and the y-values increase by 5 each time. ANSWER: 21. Jung has $200 in a safe. Each month, he adds another $10 to the safe. Miguel opens a savings account with a $200 deposit and earns 2.5% interest each month on the total amount of money in the bank. Determine if each statement is Page 18 4-7 Linear and Nonlinear Functions true or false. a. The function representing Jung’s savings is nonlinear. b. The function representing Miguel’s savings is linear. c. After 1 year, Jung will have saved $320. SOLUTION: Create a table with three columns, one to represent the number of months, m, one for Jung’s total savings, and one for Miguel’s total savings. Jung starts with $200 and increases his savings by $10 a month. Miguel starts with $200 but he earns 10% interest on his money each month. Compare the rates of change. Jung’s savings represent a linear function because the rate of change is constant. As the months increase by 1, the savings increase by 10. Miguel’s savings represent a nonlinear function because as the months increase by 1, the total savings increases at a growing rate. Jung's account can be represented by using slope-intercept form, y = 10x + 200, where x represents the number of months and y represents the amount in the savings account. Since there are 12 months in a year, the amount in Jung's account after one year can be found by replacing x with 12. y = 10(12) + 200 = 120 + 200 = 320 After one year there will be $320 in Jung's account. ANSWER: a. False b. False Page 19 4-7 Linear and Nonlinear Functions c. True Find each function value. 22. f(5) if f(x) = 3x + 4 SOLUTION: Substitute 5 for x into the function rule and evaluate. ANSWER: 19 23. f(–3) if f(x) = 2x – 8 SOLUTION: Substitute –3 for x into the function rule and evaluate. ANSWER: –14 24. f(7) if f(x) = 9x – 24 SOLUTION: Substitute 7 for x into the function rule and evaluate. ANSWER: 39 Page 20 4-7 Linear and Nonlinear Functions 25. The table shows the average number of phone calls Riley makes per day. a. Write an equation to find the total number of phone calls made in any number of days. Describe the relationship in words. b. Use the equation to determine how many phone calls Riley would make in 1 week. SOLUTION: a. Use the table to analyze the pattern. The pattern is 5 calls per day. Let d represent the number of days and c represent the number of calls. The equation is c = 5d. Riley makes an average of 5 phones calls per day. b. Write the equation. Replace d with 7. Multiply. c = 5d c = 5(7) c = 35 Riley would make 35 phone calls in one week. ANSWER: a. c = 5d; Riley makes an average of 5 phone calls per day. b. 35 phone calls Page 21