Direct Variation

Vocabulary Direct Variation Constant of variation for a direct variation

Direct Variation A recipe for paella calls for 1 cup of rice to make 5 servings. In other words, a chef needs 1 cup of rice for every 5 servings. The equation y = 5x describes this relationship. In this relationship, the number of servings varies directly with the number of cups of rice.

Definition Direct Variation – a special type of linear relationship that can represented by a function in the form y = kx, where k ≠ 0. Constant of Variation – is k, the coefficient of x, in the function y = kx. By dividing each side of y = kx by x, you can see that the ratio of the variables is constant: y/x = k. Direct Variation Equations ALWAYS go through the origin!!! Constant of Variation = Slope, so k = m!

Direct Variation

Direct Variation

Determining Direct Variation To determine whether an equation represents a direct variation; Given an equation. Solve for y. If you can write the equation in the form y = kx, where k ≠ 0, then it is a direct variation. Given a table of values. Find y/x for each ordered pair. If the ratio y/x is constant for all ordered pairs, then it is a direct variation. Given a graph. The graph is a line and passes through the origin, then it is a direct variation. The slope of the line is k.

Example: Determine Direct Variation from a Table Tell whether the relationship is a direct variation. Explain. Find for each ordered pair. This is a direct variation because is the same for each ordered pair.

Ex. 1 Graph y = 4x Write the slope as a ratio. 4 = 4 1 Graph (0,0) Use slope to find next point. Draw line through points 1 4

Example: Determine Direct Variation from a Table Tell whether the relationship is a direct variation. Explain. Find for each ordered pair. … This is not direct variation because is the not the same for all ordered pairs.

Example: Writing a Direct Variation Equation The value of y varies directly with x, and y = 3, when x = 9. Find y when x = 21. Find the value of k and then write the equation. y = kx Write the equation for a direct variation. 3 = k(9) Substitute 3 for y and 9 for x. Solve for k. Since k is multiplied by 9, divide both sides by 9. The equation is y = x. When x = 21, y = (21) = 7.

Example: Graphing Direct Variation A group of people are tubing down a river at an average speed of 2 mi/h. Write a direct variation equation that gives the number of miles y that the people will float in x hours. Then graph. Step 1 Write a direct variation equation. distance = 2 mi/h times hours y 2  x

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