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2-3: Direct Variation

2-3: Direct Variation. Essential Question: What distinguishes a direct variation from a typical function?. 2-3: Direct Variation. A linear function defined by the form y = kx, where k ≠ 0 represents direct variation . As with any line, k (the number before the x) represents the slope

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2-3: Direct Variation

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  1. 2-3: Direct Variation Essential Question: What distinguishes a direct variation from a typical function?

  2. 2-3: Direct Variation • A linear function defined by the form y = kx, where k ≠ 0represents direct variation. • As with any line, k (the number before the x) represents the slope • The direct variation is special because the y-intercept is always 0, meaning the graph goes through the origin [the point (0,0)] • When x and y are variables, you can write k = y/x. • Proof: • y = kxx xy/x = k • k is called the constant of variation.

  3. 2-3: Direct Variation • Example 1: Identifying direct variation from a table. • For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. • Because every y/x gives the Because the ratios are differentsame answer, this is a direct y/x is not constant, and this is NOTvariation. a direct variation.The constant of variation is 4.The equation is y = 4x. 8/2 = 4 12/3 = 4 20/5 = 4 4/1 = 4.0 7/2 = 3.5 16/5 = 3.2

  4. 2-3: Direct Variation • Your Turn: Identifying direct variation from a table. • Which of the following functions are direct variations? For the ones that are, find the constant of variation and write the equation -2/-1 = 2 4/3 = 1 1`/3 7/6 = 1 1/6 Not a DV 5/-9 = -5/9 1 2/3 3 = 5/9 3 5/8 6 = 29/48 Not a DV -2/-6 = 1/3 1/3 = 1/3 4/12 = 1/3 k = 1/3 y = 1/3 x

  5. 2-3: Direct Variation • Example 2: Identifying direct variation from an equation. • For each function, determine whether y varies directly with x. If so, find the constant of variation. • This is a direct variation This is not a direct variationThe constant of variation is 2/3. y = 2x + 3 This is not in the form y = kx 3y = 2x 3 3 y = 2/3 x

  6. 2-3: Direct Variation • Your Turn: Identifying direct variation from an equation. • Which of the following functions are direct variations? For the ones that are, find the constant of variation and write the equation. • y = x/2 • 2y – 1 = x • 5/6 x = 1/3 y • 7x + 4y = 10 Is a direct variation k = ½ Not a direct variation Is a direct variation (divide both sides by 1/3) k = 5/2 Not a direct variation

  7. 2-3: Direct Variation • Example 3: Finding additional values. • For each function, find the constant of variation.Then find the value of y when x = -5 y = -5 when x = 3 Use the equation: y = kx -5 = k(3) 3 3 -5/3 = k When x is -5: y = kx y = (-5/3)(-5) y = 25/3 y = -2/3 when x = -1/3 Use the equation: y = kx -2/3 = k(-1/3) -1/3 -1/3 2 = k When x is -5: y = kx y = (2)(-5) y = -10

  8. 2-3: Direct Variation • Your Turn: Finding additional values. • For each function, find the constant of variation.Then find the value of y when x = -5 y = ½ when x = -2 Use the equation: y = kx ½ = k(-2) -2 -2 -1/4 = k When x is -5: y = kx y = (-1/4)(-5) y = 5/4

  9. 2-3: Direct Variation • Because direct variations have the same constant of variation, the ratio of y/x always remains the same. • This means unknowns can also be found by using proportions. • Example 4: Using a proportion • Suppose y varies directly with x, and x = 27 when y = -51.Find x when y = -17. Let (x1, y1) = (27, -51) and let (x2, y2) = (x2, -17)

  10. 2-3: Direct Variation • Your Turn: Using a proportion • Suppose y varies directly with x, and y = -7 when x = 2.Find y when x = 8. Let (x1, y1) = (2, -7) and let (x2, y2) = (8, y2)

  11. 2-3: Direct Variation • Assignment • Page 74 – 75 • Problems 1 – 27 • Odd problems only • Skip #23

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