PARTIAL and DIRECT VARIATION

1. PARTIAL and DIRECT VARIATION

2. Recall: If a relationship is linear, its graph forms a straight line.

3. Recall: If a relationship is linear, its graph forms a straight line.

4. We can further divide linear relationships into two categories: • direct linear relationships • partial linear relationships

5. DIRECT PARTIAL • when the independent variable is zero, the dependent variable is also zero • both sides of the equation have only one term ex: x = 0 y = 2x y = 2(0) y = 0

6. DIRECT PARTIAL • when the independent variable is zero, the dependent variable is also zero • both sides of the equation have only one term ex: x = 0 y = 2x y = 2(0) y = 0 • when the independent variable is zero, the dependent variable is not zero • one side of the equation has two terms • ex: x = 0 y = 2x + 5 • y = 2(0) + 5 • y = 5

7. DIRECT PARTIAL • value of the dependent variable is based solely on the value of the independent variable • passes through the origin (0,0)

8. DIRECT PARTIAL • value of the dependent variable is based solely on the value of the independent variable • passes through the origin (0,0) • value of the dependent variable is based on both the independent variable and a constant • does not pass through the origin

9. Which of the following lines show: • Partial Variation? • Direct Variation? A B C

10. Which of the following lines show: • Partial Variation? B • Direct Variation? A,C A B C

11. y = 3x • C = 200 + 4d • a = – 70 + b • h = 5t • Which of the equations are partial linear relationships? • Which of the equations are direct linear relationships?

12. y = 3x • C = 200 + 4d • a = – 70 + b • h = 5t • Which of the equations are partial linear relationships? • 2,3 • Which of the equations are direct linear relationships? • 1,4

13. We can’t use x and y intercepts alone to graph a direct linear relationship. ex: y = 3x x-intercept: y = 0 0 = 3x 0 = x (0,0) y-intercept: x = 0 y = 3(0) y = 0 (0,0) The x and y intercepts are the same, and we need two points to graph a line.

14. Calculating Steepness with Rate Triangles

15. We can also compare linear relationships based on their steepness. We determine the steepness of a line using a rate triangle. hypotenuse height base

16. Using any two points on the line as the hypotenuse, draw a right angled triangle. The rate is a measure of steepness where: rate = height base 6 3

17. Using any two points on the line as the hypotenuse, draw a right angled triangle. The rate is a measure of steepness where: rate = height base = 2 6 3

18. Determine the rate/steepness of lines A, B, and C. A B C

19. Determine the rate/steepness of lines A, B, and C. A 25 B C 5

20. Determine the rate/steepness of lines A, B, and C. A B 5 C 5

21. Determine the rate/steepness of lines A, B, and C. A B C 10 5

22. Determine the rate/steepness of lines A, B, and C. A. r = 25 = 5 B. r = 2 = 1 C. r = 10 = 2 5 2 5

23. Interpolation And Extrapolation

24. We can use the fact that a relationship is linear to identify other data points without performing calculations. • interpolation: finding another data point that exists between two points you already know

25. We can use the fact that a relationship is linear to identify other data points without performing calculations. • interpolation: finding another data point that exists between two points you already know • extrapolation: finding another point that exists beyond the points you already know (extend the line)

26. INTERPOLATION EXTRAPOLATION