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Direct Variation, and Proportion Lesson 1.4

Direct Variation, and Proportion Lesson 1.4. Objectives. Write and apply direct-variation equations. Write and solve proportions. Determine if the values in a table, graph, or equation represent direct variation. Glossary Terms. constant of variation

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Direct Variation, and Proportion Lesson 1.4

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  1. Direct Variation, and Proportion Lesson 1.4

  2. Objectives • Write and apply direct-variation equations. • Write and solve proportions. • Determine if the values in a table, graph, or equation represent direct variation. Glossary Terms • constant of variation • Cross-Product Property of Proportions • direct variation • proportion

  3. Each day Jonathan rides his bicycle for exercise. When traveling at a constant rate, he rides 4 miles in about 20 minutes. At this rate, how many miles would he travel in 35 minutes? Make a table of values min 20 25 30 35 miles 4 5 6 7

  4. How is the distance (d) traveled in miles related to the time (t) in minutes? Write an equation. d = 1/5t Graph the equation. 10 miles How long would it take Jonathan to travel 10 miles? 0 50 min 50 minutes

  5. What features of the graph and equation are typical for anyone traveling at a constant rate? The shape is linear. The graph passes through (0,0). The equation is of the form y = kx where k is the constant rate. d = rt; r is the constant rate

  6. This is an example of direct variation. The time is proportional to the distance traveled. The equation for direct variation is in the form y = kx, where k is the constant of variation. The constant of variation represents the rate of change of y with respect to x. k = y/x.

  7. k = 27 5 1.4 Direct Variation and Proportion Key Skills Write a direct-variation equation for given variables. Find the direct-variation equation for y = 27 when x = 5. = 5.4 y = 5.4x

  8. Suppose that when Jonathan is riding, he travels 5 miles in about 30 minutes. Write and solve an equation to determine long it would take him to travel 12 miles. k = 5/30 = 1/6 (mi/min) d = (1/6)t 12 = (1/6)t t = 72 minutes

  9. Jack planted flowers. After 2 weeks the plants measured 3 cm tall. Give the plant’s height (y) in cm x weeks after sprouting. Y = 1.5x This is an example of direct variation. 1.5 is the rate of change.

  10. If Maria buys the plant when it is 4 cm tall and still growing 1.5 cm/week, then what is the plant’s height x weeks after she buys it? Y = 1.5x + 4 This is no longer direct variation. The height is not directly related to the number of weeks.

  11. Make a table showing the number of weeks (x) and the height of the plant f(x). X 0 1 2 3 4 f(x) 4 5.5 7 8.5 10 Write the function: f(x) = 1.5x + 4

  12. Graph the function Notice that although this graph has the same rate of change (1.5), it does not pass through (0,0). This is why it is not direct variation. 10 8 6 4 2 Height (cm) 0 1 2 3 4 5 Time (weeks)

  13. TO DETERMINE IF A LINEAR FUNCTION IS DIRECT VARIATION: Put the equation in y = mx + b form. If b = 0, it is direct. (y = kx) TO DETERMINE IF A TABLE SHOWS DIRECT VARIATION: If every y/x entry gives the same value, it is direct. TO DETERMINE IF A GRAPH SHOWS DIRECT VARIATION: If the graph of the line goes through the origin, it is direct variation.

  14. Decide whether each function is linear. If it is, state whether it represents direct variation. a. y = 2x + 4 Linear, not direct Not linear b. y = 3x2 Linear, direct c. f(x) = -3x Linear, direct d. y = (2/3)x

  15. Proportion Property of Direct Variation If (x1,y1) and (x2,y2) satisfy y = kx, then

  16. Exploring Similarity andDirect Variation Do the activity on p. 31. Does similarity in similar triangles indicate a direct-variation relationship between the lengths of the sides of the similar triangles?

  17. If y varies directly as x, then y is proportional to x. A proportion is a statement that two ratios are equal. A ratio is the comparison of two quantities by division such as Proportions can be solved by using cross products.

  18. 35 87.5 = 4 x 35 87.5 35 x 350 4 x 1.4 Direct Variation and Proportion Key Skills Write and solve proportions. 35x = 350 x = 10

  19. Solve: 5x = 6x - 2 x = 2

  20. Solve: 7x = 6x + 4 x = 4 Solve: 14a - 7 = 15a -7 = a

  21. Newton’s Law of Gravitation Find the weight of the Sojourner rover on Mars (w) if it weighs 24.3 pounds on earth. 100w=923.4 w = 9.23 lb

  22. Write a direct-variation equation that gives the weight of an object on Mars (m) in terms of its weight on earth (e) m/e = 38/100 m = .38e

  23. Graph the equation and verify that a weight of 24.3 lb on earth is equal to 9.2 lb on Mars.

  24. If a > 0, how many solutions does have? Find the solutions and justify your answer. x2 = a2 x =  a two solutions

  25. Homework: p. 33 (19–25 odd, 33 – 45 odd, 52-55, 60, 64, 66, 67, 71 The End

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