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Unit 3, Lesson 6 Constant of Proportionality and Writing Direct Variation Equations
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Proportional relationships vary directly and we say that they are direct variations. These are linear (line) relationships and have an equation that describes the relationship. The constant of proportionality in a direct variation is a constant ratio (unit rate) in any proportional relationship. We use the letter k to represent the constant of proportionality. Equations: y = kx or k = y x
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We can find the constant of proportionality from a table of values, equation and a graph. In a table, simplify any one of the ratios. Chaperones12345 Students1224364860
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Apples (lbs)22.533.54 Cost ($)3.964.955.946.937.92 Example: Find the constant of proportionality:
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XY 21.5 53.75 107.5 129 Find the constant of proportionality.
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In an equation, write the equation in the form y = kx to find k. Examples: Y=5x Y= ¼ x Y=3.5x
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Find the constant of proportionality: (click to reveal)
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In a graph, choose a point (x, y) to find and simplify the ratio to find k. (2, 24) Chaperones Students 0 1 2 3 4 5 6 7 8 9 10 6 12 18 24 30 36 42 48 54 60
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Example: Find the constant of proportionality. 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 4 8 12 16 20 24 28 32 36 40 Find the constant of proportionality.
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Constant of proportionality & unit rate are equivalent. We can use the constant of proportionality to help write equations using proportional relationships. By transforming the equation from: to y = kx, we can write an equation that can be applied to various situations. X is the independent (input) variable and y is the dependent (output) variable. This means that a change in x will effect y.
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Example: You are buying Jersey Tomatoes for a cost of 2 pounds for $3.98. Write an equation to represent the proportional relationship. Let c = cost p = pounds Determine the unit rate: Write an equation to relate the two quantities:
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Example: At the candy store, you purchase 5 lbs for $22.45. Write an equation to represent the proportional relationship. Let c = cost p = pounds Determine the unit rate: Write an equation to relate the two quantities:
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c = 1.4p p = 1.4c Write an equation that represents the proportional relationship. The total cost (c) of grapes for $1.40 per pound(p) A B
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0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Write an equation that represents the proportional relationship. A B C D
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Days, d2345 Hours, h1725.53442.5 Write an equation that represents the proportional relationship. h = 8.5 d A B C D d = 8.5 h
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