Chapter 1.9 Direct Variation

Vocab 𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥 𝑘≠0 direct variation constant of variation When two variable quantities have a constant ratio, their relationship is called a _____________________. The constant ratio is called the ______________________. Constant of variation = constant of _________________. 𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥 𝑘≠0 direct variation constant of variation proportionality

The pool fills at a rate of 0.4 inches per minute Example 1 The height of the water as a pool is being filled is shown in the graph. Determine the rate in inches per minute. The pool fills at a rate of 0.4 inches per minute ℎ𝑒𝑖𝑔ℎ𝑡 𝑡𝑖𝑚𝑒 = 𝑦 𝑥 = 2 5 4 10 6 15 8 20 0.4 0.4 0.4 0.4

Two minutes after a diver enters the water, he descends 52 feet Two minutes after a diver enters the water, he descends 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the scuba diver descending? 𝑓𝑒𝑒𝑡 𝑚𝑖𝑛𝑢𝑡𝑒 = 52 2 130 5 𝑦 𝑥 = OR 26 OR 26 The scuba diver is descending 26 feet per minute

The constant of proportionality is 10. So, Julie earns $10 per hour. Example 2 The equation 𝑦=10𝑥 represents the amount of money 𝑦 that Julie earns for 𝑥 hours of work. Identify the constant of proportionality. Explain what is represents in this situation. 𝑦=𝑘𝑥 𝑦=10𝑥 The constant of proportionality is 10. So, Julie earns $10 per hour.

Got it? The distance 𝑦 traveled in miles by the Chang family in 𝑥 hours is represented by the equation 𝑦=55𝑥. Identify the constant of proportionality. Then explain what it represents in this situation. 𝑦=𝑘𝑥 𝑦=55𝑥 The constant of proportionality is 55. So, The Chang family traveled 55 miles per hour.

Determine Direct Variation Not all situations with a constant rate of change are proportional relationships. Not all linear functions are direct variations.

Example 3 There is no constant ratio and the line does not go through the origin. So, there is no direct variation. Pizza costs $8 plus a $3 delivery charge. Show the cost of 1, 2, 3, and 4 pizzas. Is there a direct variation? Number of pizzas 1 2 3 4 Cost ($) 𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑧𝑧𝑎𝑠 = $11 $19 $27 $35 11 1 19 2 27 3 35 4 =11 =9.5 =9 =8.75

Two pounds of cheese cost $8. 40 Two pounds of cheese cost $8.40. Show the cost of 1, 2, 3, and 4 pounds of cheese. Is there a direct variation? Explain. Pounds of cheese 1 2 3 4 Cost ($) 𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑢𝑛𝑑𝑠 =

Example 4 Since the ratios are the same, the relationship is a direct variation. The constant of proportionality is 12. Determine whether the linear relationship is a direct variation. If so, state the constant of proportionality. 𝑤𝑎𝑔𝑒𝑠 𝑡𝑖𝑚𝑒 = 12 1 24 2 36 3 48 4 =12 =12 =12 =12

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Chapter 1.9 Direct Variation

Vocab When two variable quantities have a constant ratio, their relationship is called a _____________________. The constant ratio is called the ______________________. Constant of variation = constant of _________________. 𝑦 𝑥 =𝑘 𝐎𝐑 𝑦=𝑘𝑥 𝑘≠0

Example 1 The height of the water as a pool is being filled is shown in the graph. Determine the rate in inches per minute. ℎ𝑒𝑖𝑔ℎ𝑡 𝑡𝑖𝑚𝑒 = 𝑦 𝑥 =

Two minutes after a diver enters the water, he descends 52 feet Two minutes after a diver enters the water, he descends 52 feet. After 5 minutes, he has descended 130 feet. At what rate is the scuba diver descending? 𝑓𝑒𝑒𝑡 𝑚𝑖𝑛𝑢𝑡𝑒 = 𝑦 𝑥 = The scuba diver is descending ____________________.

Example 2 The equation 𝑦=10𝑥 represents the amount of money 𝑦 that Julie earns for 𝑥 hours of work. Identify the constant of proportionality. Explain what is represents in this situation. 𝑦=𝑘𝑥 The constant of proportionality is ______. So, Julie earns ______________________.

Got it? The distance 𝑦 traveled in miles by the Chang family in 𝑥 hours is represented by the equation 𝑦=55𝑥. Identify the constant of proportionality. Then explain what it represents in this situation. 𝑦=𝑘𝑥 The constant of proportionality is ________. So, The Chang family traveled ________________.

Determine Direct Variation Not all situations with a constant rate of change are proportional relationships. Not all linear functions are direct variations.

Example 3 Pizza costs $8 plus a $3 delivery charge. Show the cost of 1, 2, 3, and 4 pizzas. Is there a direct variation? Number of pizzas 1 2 3 4 Cost ($) 𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑖𝑧𝑧𝑎𝑠 =

Two pounds of cheese cost $8. 40 Two pounds of cheese cost $8.40. Show the cost of 1, 2, 3, and 4 pounds of cheese. Is there a direct variation? Explain. Pounds of cheese 1 2 3 4 Cost ($) 𝑐𝑜𝑠𝑡 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑜𝑢𝑛𝑑𝑠 =

Example 4 Determine whether the linear relationship is a direct variation. If so, state the constant of proportionality. 𝑤𝑎𝑔𝑒𝑠 𝑡𝑖𝑚𝑒 =

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