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Direct Variation Section 5-2 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 • ________________– a special type of linear relationship that can be represented by a function in the form y = kx, where k0. • ________________– 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 Direct Variation Determining Direct Variation • To determine whether an equation represents a direct variation; 1) Given an equation. 2) Given a table of values. 3) Solve for y. If you can write the equation in the form y = kx, where k 0, then it is a direct variation. 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 an Equation Tell whether the equation represents a direct variation. If so, identify the constant of variation. y = 3x This equation ___________ a direct variation because it is in the form of y = kx. The constant of variation is ______. Example: Determine Direct Variation from an Equation Tell whether the equation represents a direct variation. If so, identify the constant of variation. 3x + y = 8 –3x –3x y = –3x + 8 Solve the equation for y. Since 3x is added to y, subtract 3x from both sides. This equation ____________a direct variation because it ___________ be written in the form y = kx. Example: Determine Direct Variation from an Equation Tell whether the equation represents a direct variation. If so, identify the constant of variation. –4x + 3y = 0 This equation ____________ direct variation because it ______ in the form of y = kx. The constant of variation is _____ . Your Turn: Tell whether the equation represents a direct variation. If so, identify the constant of variation. 3y = 4x + 1 This equation ___________a direct variation because it _________written in the form y = kx. Your Turn: Tell whether the equation represents a direct variation. If so, identify the constant of variation. Solve the equation for y. 3x = –4y This equation represents a direct variation because it is in the form of y = kx. The constant of variation is . Your Turn: Tell whether the equation represents a direct variation. If so, identify the constant of variation. y + 3x = 0 This equation ____________ a direct variation because it is ______in the form of y = kx. The constant of variation is _____. Determine Direct Variation from a Table What happens if you solve y = kxfor k? y = kx Divide both sides by x (x ≠ 0). So, in a direct variation, the ratio is equal to the constant of variation. Another way to identify a direct variation is to check whether is the same for each ordered pair (except where x = 0). 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 pair. is the same for each ordered 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 ordered pairs. is the not the same for all Your Turn: Tell whether the relationship is a direct variation. Explain. Find for each ordered pair. This __________direct variation becausey/x is ________the same for all ordered pairs. Your Turn: Tell whether the relationship is a direct variation. Explain. Find for each ordered pair. This _________ a direct variation becauset _______ the same for each ordered pair. 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. Your Turn: The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10. Find the value of k and then write the equation. y = kx Write the equation for a direct variation. The equation is y =___. When x = 10, y==_____. 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 y = = 2 mi/h 2 times hours • x Example: Continued Step 2 Choose values of x and generate ordered pairs. x y = 2x (x, y) 0 y = 2(0) = 0 (0,0) 1 y = 2(1) = 2 (1,2) 2 y = 2(2) = 4 (2,4) Step 3 Graph the points and connect. Your Turn: The perimeter y of a square varies directly with its side length x. Write a direct variation equation for this relationship. Then graph. Step 1 Write a direct variation equation. perimeter = ___sides y = ___ times • length x Your Turn: Continued Step 2 Choose values of x and generate ordered pairs. x y = 4x (x, y) 0 (0,) 1 (1,) 2 (2,) Your Turn: Continued Step 3 Graph the points and connect. Joke Time • What did the horse say when he got to the bottom of his feed bag?