Algebra 4-5 Study Guide: Direct Variation (pp ) Page! 1 of! 9

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1 Page! 1 of! 9 Attendance Problems. Solve for y y = 2x 2. 6x = 3y 3. Write an equation that describes the relationship. Solve for x. 3 4.! 5.! 5 = x = 1.5 x I can identify, write, and graph direct variation. Vocabulary direct variation constant of variation Common Core CC.9-12.A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.* CC.9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. A direct variation is a special type of linear relationship that can be written in the form y = kx, where k is a nonzero constant called the constant of variation.

2 Page! 2 of! 9 Video Example 1. Tell whether the equation represents a direct variation. If so, identify the constant of variation. A. y = 5x B) 2x + 3y = 0 C) 4x + y = 8 1 Identifying Direct Variations from Equations Tell whether each equation represents a direct variation. If so, identify the constant of variation. A y = 4x This equation represents a direct variation because it is in the form y = kx. The constant of variation is 4. B -3x + 5y = 0-3x + 5y = 0 Solve the equation for y. + 3x + 3x Since -3x is added to 5y, add 3x to both sides. 5y = 3x _ 5y _ Since y is multiplied by 5, divide both sides by 5. 5 = 3x 5 y = 3_ 5 x This equation represents a direct variation because it can be written in the form y = kx. The constant of variation is 3 5. C 2x + y = 10 2x + y = 10 Solve the equation for y. - 2x - 2x Since 2x is added to y, subtract 2x from both sides. y = -2x + 10 This equation does not represent a direct variation because it cannot be written in the form y = kx. Parent: How is your math homework going? Child: Well, m is negative, so at least I m on the down slope!

3 Page! 3 of! 9 Example 1. Tell whether the equation represents a direct variation. If so, identify the constant of variation. A. y = 3x B. 3x + y = 8 C. -4x + 3y = 0 Guided Practice. Tell whether the equation represents a direct variation. If so, identify the constant of variation. 6. 3y = 4x x = -4y 8. y + 3x = 0 Desmos Activity: Constant of Proportionality: 5tg8 9. Solve y = kx for k. Video Example 2. Tell whether the relationship is a direct variation. Explain. A) B)!!

4 Page! 4 of! 9 2 Identifying Direct Variations from Ordered Pairs Tell whether each relationship is a direct variation. Explain. A x y Method 1 Write an equation. y = 6x Each y-value is 6 times the corresponding x-value. This is a direct variation because it can be written as y = kx, where k = 6. Method 2 Find y x for each ordered pair. 6_ 1 = 6 _ 18 3 = 6 30_ 5 = 6 This is a direct variation because y x B x y Method 1 Write an equation. y = x - 4 is the same for each ordered pair. Each y-value is 4 less than the corresponding x-value. This is not a direct variation because it cannot be written as y = kx. Method 2 Find y x for each ordered pair. _-2 2 = -1 0_ 4 = 0 4_ 1_ 8 = 2 This is not a direct variation because y x is not the same for all ordered pairs. Stupidity, like virtue, is its own reward.

5 Page! 5 of! 9 Example 2. Tell whether the relationship is a direct variation. Explain. A) B)!! Guided Practice. Tell whether the relationship is a direct variation. Explain !!! Video Example 3. The value of y directly with x, and y = 6 and x = 30. Find y when x = 45.

6 Page! 6 of! 9 3 Writing and Solving Direct Variation Equations The value of y varies directly with x, and y = 6 when x = 12. Find y when x = 27. Method 1 Find the value of k and then write the equation. y = k x 6 = k (12) Write the equation for a direct variation. Substitute 6 for y and 12 for x. Solve for k. 1_ Since k is multiplied by 12, divide both sides by = k The equation is y = _ 1 2 x. When x = 27, y = _ 1 (27) = Method 2 Use a proportion. 6_ 12 = _ y In a direct variation, _ y 27 x is the same for all values of x and y. 12y = 162 Use cross products. y = 13.5 Since y is multiplied by 12, divide both sides by 12. Example 3. The value of y varies directly with x, and y = 3, when x = 9. Find y when x = Guided Practice. The value of y varies directly with x, and y = 4.5 when x = 0.5. Find y when x = 10. (pp 263) 11,

7 Page! 7 of! 9 Video Example 4. The Australian Tiger Beetle is on e of the fastest land insects. One the ground, it can travel at a speed of about 2.5 meters per second. Write a direct variation equation for the distance y a beetle will travel in x seconds. Then graph. EXAMPLE 4 Graphing Direct Variations The three-toed sloth is an extremely slow animal. On the ground, it travels at a speed of about 6 feet per minute. Write a direct variation equation for the distance y a sloth will travel in x minutes. Then graph. Step 1 Write a direct variation equation. distance = 6 feet per minute times number of minutes y = 6 x Step 2 Choose values of x and generate ordered pairs. Step 3 Graph the points and connect. x y = 6x (x, y) 0 y = 6 (0) = 0 (0, 0) 1 y = 6 (1) = 6 (1, 6) 2 y = 6 (2) = 12 (2, 12)

8 Page! 8 of! 9 Example 4. 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.!

9 ! Algebra 4-5 Study Guide: Direct Variation (pp ) Page! 9 of! Guided Practice. The perimeter y of a square varies directly with its side length x. Write a direct variation equation for this relationship. Then graph. 4-5 Direct Variation Desmos Activity: Constant of Proportionality: 5tg8 (p 263) 11, 13-17, 19, 20, 22, A Ready to Go On pretest & posttests.