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## PARTIAL and DIRECT VARIATION

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**Recall: If a relationship is linear, its graph forms a**straight line.**Recall: If a relationship is linear, its graph forms a**straight line.**We can further divide linear relationships into two**categories: • direct linear relationships • partial linear relationships**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**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**DIRECT PARTIAL**• value of the dependent variable is based solely on the value of the independent variable • passes through the origin (0,0)**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**Which of the following**lines show: • Partial Variation? • Direct Variation? A B C**Which of the following**lines show: • Partial Variation? B • Direct Variation? A,C A B C**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?**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**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.**We can also compare linear relationships based on their**steepness. We determine the steepness of a line using a rate triangle. hypotenuse height base**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**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**Determine the rate/steepness of lines A, B, and C.**A 25 B C 5**Determine the rate/steepness of lines A, B, and C.**A B 5 C 5**Determine the rate/steepness of lines A, B, and C.**A B C 10 5**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**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**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)