DIRECT VARIATION TABLES AND SLOPE LESSON 3-B When a graph on a coordinate plane is a straight line that goes through the origin it is called a direct variation graph. In this lesson you will investigate direct variation tables and graphs in Quadrant I. As the -coordinate gets larger, the y-coordinate also gets larger at a constant rate. For eample, Stein walks 4 miles per hour in a race. Hours, Miles, y 1 4 2 8 3 12 4 16 5 20 Miles Hours The table shows some ordered pairs describing his time since the race started and the number of miles he has walked. For each step, the -values increase by 1 and the y-values increase by 4. The direct variation graph shows his distance from the starting line in miles over time in hours. This is a direct variation graph because it goes through the point (0, 0) and is a straight line. Cathy s job also illustrates direct variation. Cathy earns $10 per hour babysitting. If Cathy works for 0 hours, she earns $0. This means (0, 0) is a point on the coordinate plane that fits the problem. Since (0, 0) is a point on the graph and the graph increases at a constant rate of $10 per hour, a graph showing her earnings models direct variation. Money Earned ($) Hours Babysitting, Money Earned ($), y 1 10 2 20 3 30 4 40 5 50 6 60 Hours Lesson 3-B ~ Direct Variation Tables And Slope 5
You can also determine if Cathy s earnings show direct variation by looking at the table. Find the ratio of y-coordinate money earned to hours worked ( -coordinate ) for each ordered pair. If the ratios are equal, the table shows direct variation. Hours Babysitting, Money Earned, y y-coordinate -coordinate Can t divide by 0 1 10 10 1 20 2 20 2 = 10 1 30 3 30 3 = 10 1 40 4 40 4 = 10 1 50 5 50 5 = 10 1 60 6 60 6 = 10 1 For each ordered pair, the unit rate is $10 per hour, which matches Cathy s babysitting rate. The unit rate is also called the slope. The slope of a line that goes through the origin is the ratio of y _. In this situation, the slope of the line is 10. Sometimes it makes sense to connect the points on a scatter plot with a line. For eample, if Cathy works 1.5 hours, the point (1.5, 15) means that after 1.5 hours of babysitting, Cathy earned $15. Even though this point is not on the original scatter plot, if the points in the table are connected with a line, the point (1.5, 15) will be a point on the line. An equation can be written to help Cathy figure out how much money, y, she makes after hours. Look at these proportions: 10 dollars 1 hour = y dollars y = 10 3 = 30 3 hours 10 dollars 1 hour = y dollars y = 10 6 = 60 6 hours 10 dollars 1 hour = y dollars hours y = 10 Money Earned Hours Think about the ratio y_ as the constant rate of change, m. This means that: y_ = m. To write this as a function, solve for y. Multiply both sides by. y _ = m Simplify. y = m 6 Lesson 3-B ~ Direct Variation Tables And Slope
EXAMPLE 1 Use the direct variation function y = 3. a. Complete the table for the given input values. Input, 0 1 2 3 4 Function y = 3 Output, y b. Draw a scatter plot of the ordered pairs in the table and connect the ordered pairs with a straight line. c. Find the slope of the function. Solutions a. Input, Function y = 3 Output, y 0 3(0) 0 1 3(1) 3 2 3(2) 6 3 3(3) 9 4 3(4) 12 b. c. The slope is the coefficient of the variable. y = 3 Slope = 3 Also, note that y _ = 3 for each ordered pair. Lesson 3-B ~ Direct Variation Tables And Slope 7
EXAMPLE 2 Elliot wrote an equation, y = 4.5, to show the relationship between the number of days,, and the total miles, y, he ran. Jacob runs daily but only recorded his total miles run in the table at 2, 5, 10 and 11 days. Who runs more miles on average per day? Elliot Jacob y = 4.5 Days, Total Miles Run, y 2 8 5 20 10 40 11 44 Solution Elliot wrote an equation with a slope of 4.5. The slope represents his daily rate. Elliot: 4.5 miles per day Jacob recorded his mileage in a table. To find his unit rate, or slope, find the ratio of _ y. Jacob: 8 miles = 4 miles per day 2 days Elliot runs 0.5 more miles per day than Jacob. EXERCISES Each table below represents direct variation. Graph each scatter plot. Find the rate of change for each graph. 1. y 1 3 2 6 3 9 4 12 2. y 1 2 2 4 3 6 4 8 3. y 2 1 4 2 6 3 8 4 Find the slope of each direct variation graph. 4. (4, 8) 5. (3, 9) 6. (1, 2) (1, 3) (4, 2) (2, 1) 8 Lesson 3-B ~ Direct Variation Tables And Slope
7. Kirsten walked 4 miles per hour for 5 hours to train for an upcoming marathon walk. a. Copy and complete the table below. Number of hours Kirsten walked, 0 1 2 3 4 5 Number of miles Kirsten walked, y b. What is the slope of the direct variation relationship? c. How do you know that this is a direct variation relationship? 8. Julio made a pattern in art class. a. Draw the net two figures in Julio s pattern below. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 b. Complete the table showing the figure number and its corresponding number of squares. Figure number, 1 2 3 4 5 Number of squares, y c. Does this pattern model direct variation? Eplain. d. If the pattern models direct variation, find its slope. 9. Caroline went to the state fair. The table at right shows the total amount of money she spent as time passed. Tom also kept track of the amount he spent at the fair. He wrote the equation y = 10.5 to model his spending where represents hours and y represents the total amount he had spent in dollars. a. Who spent more per hour? How do you know? b. Caroline and Tom both spent 8 hours at the fair. What was the TOTAL amount of money they spent all together? Hours, Caroline Total Spent, y 1 $13 2 $26 3 $39 4 $52 10. Justin and Peter argued about who completed more homework. Justin made a graph showing his completed homework assignments. Peter made a table. a. Who completed more assignments per week? How do you know? b. In a nine week quarter, the teacher says you must complete 30 assignments to get an A for participation. Will either of the boys earn an A? Support your answer with work. Weeks, Peter s Table Total Assignments Completed, y 1 3 2 6 3 9 4 12 Lesson 3-B ~ Direct Variation Tables And Slope 9
11. Juanita opened her own movie theater. She plans to show older movies at a reduced price compared to the large theater in town that shows new releases. She plans to charge $3.00 per person and hopes to fill her 50-seat theater once in the late afternoon and once in the evening. a. Complete this table to show how much money Juanita will get for selling the given numbers of tickets to a show. Number of tickets sold, 0 10 15 20 40 50 Money collected ($), y b. Can this situation be modeled by direct variation? Eplain why or why not. c. What is the slope of this relationship? d. Write an equation for the amount of money collected (y) based on the number of tickets sold (). e. If Juanita sells out both shows each day, how much money will she collect? 12. Bryan and Megan tried to write an equation for the direct variation relationship given in the table below. Each student used a different method. 0 4 8 12 16 20 y 0 1 2 3 4 5 Look at both solutions and determine who is correct. For the student that made a mistake, identify the mistake and eplain what the student should have done instead to use their method correctly. Bryan s Work Use the point (4, 1) to find the constant 4 1 = 4 rate of change. Megan s Work Use the point (4, 1) to set up a proportion 1 4 = _ y to find the equation. The equation is: y = 4 Use cross products. 4y = Divide both sides by 4. y = 4 The equation is: y = 1 4 13. Use the figures to answer the questions. a. Draw the net two figures in the pattern below. Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 b. Complete the table showing the figure number and its corresponding number of squares. Figure number, 1 2 3 4 5 6 7 Number of squares, y c. Does the table show direct variation? Eplain why or why not. d. Write an equation which gives the number of squares in the pattern based on the figure number. e. Use your equation to determine how many squares would be in Figure 25. 10 Lesson 3-B ~ Direct Variation Tables And Slope