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Understanding Direct Variation: Slope, Equations, and Graphs in Linear Relationships

This homework review focuses on the concept of direct variation in mathematics, highlighting its definition as a linear relationship between two variables represented by the equation y = kx, where k is the constant of variation (slope). Students will learn how to identify direct variation from equations, interpret graphs, and solve for unknowns. Specific problems are provided, including determining the value of y from given points and analyzing whether graphs represent direct variations by checking if they pass through the origin.

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Understanding Direct Variation: Slope, Equations, and Graphs in Linear Relationships

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  1. 5 – 4 A: Direct Variation

  2. Homework Review A line has a slope of 5 and passes through the points (4,3) and (2,y). What is the value of y? y = -7

  3. Homework Review A line passes through the origin and has a slope of . Through which quadrants does the line pass? II and IV

  4. Definitions Direct variation – linear relationship between two variables that can be written in the form y = kx Constant of variation – the fixed number (k) in a direct variation (the coefficient) **This is another expression that means slope or rate of change.***

  5. Direct Variation • Will be a straight line when graphed • ALWAYS passes through the origin (0,0)

  6. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y = 2x Yes k = 2

  7. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y = 1/3 x Yes k = 1/3

  8. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y = - ½x Yes k = -½

  9. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y = 2x + 3 No

  10. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y = ½x - 6 No

  11. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 2y = x

  12. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. 2y = x 2 2 y = ½ x Yes, this is a direct variation. ½ is the constant of variation.

  13. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y + 1 = 2x .

  14. Identifying a Direct Variation from an Equation Tell whether each equation represents a direct variation. If so, identify the constant of variation. y + 1 = 2x Solve for y. - 1 - 1 . y = 2x – 1 . This is not in the form y = kx, so this is not a direct variation.

  15. Work with your partner. Page 214 (1-4)

  16. Notes: The graph of a direct variation is a line that passes through the origin (0 0). The constant of variation (k) is the slope of the line.

  17. Does this graph represent a direct variation? • Yes, the line passes through the origin, so this is a direct variation. • What is the slope of the line (constant of variation)? k = 1

  18. Does this graph represent a direct variation? • Yes, the line passes through the origin, so this is a direct variation. • What is the constant of variation? k = -½

  19. Does this graph represent a direct variation? • No, the line does not pass through the origin, so this is NOT a direct variation.

  20. Does this graph represent a direct variation? • No, the line does not pass through the origin, so this is NOT a direct variation.

  21. Does this graph represent a direct variation? • Yes, the line passes through the origin, so this is a direct variation. • What is the constant of variation? k = 2 y = 2x 22 20 18 16 14 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10

  22. Does this graph represent a direct variation? • No, this is not a straight line, so this is NOT a direct variation.

  23. Let’ look at: Page 214 (7-8)

  24. Partner Talk Page 215 (19-24) Homework: Page 214 (10-13, 16-17)

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