Direct Variation PDF

Direct Variation What is it and how do I know when I see it? Definition: Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx) Another way of writing this is k = In other words: * the constant of variation (k)...

Direct Variation What is it and how do I know when I see it? Definition: Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx) Another way of writing this is k = In other words: * the constant of variation (k) in a direct variation is the constant (unchanged) ratio of two variable quantities.  Examples of Direct Variation: Note: X increases, 6,7,8 And Y increases. 12, 14, 16 What is the constant of variation of the table above? Since y = kx we can say Therefore: 12/6=k or k = 2 14/7=k or k = 2 y = 2x is the 16/8=k or k =2 Note k stays constant. equation!  Examples of Direct Variation: Note: X decreases, 10, 5, 3 And Y decreases. 30, 15, 9 What is the constant of variation of the table above? Since y = kx we can say Therefore: 30/10=k or k = 3 15/5=k or k = 3 y = 3x is the 9/3=k or k =3 Note k stays constant. equation!  Examples of Direct Variation: Note: X decreases, -4, -16, -40 And Y decreases. -1,-4,-10 What is the constant of variation of the table above? Since y = kx we can say Therefore: -1/-4=k or k = ¼ -4/-16=k or k = ¼ y = ¼ x is the equation! -10/-40=k or k = ¼ Note k stays constant. What is the constant of variation for the following direct variation? 1. 2 2. -2 3. -½ 4. ½  Is this a direct variation? If yes, give the constant of variation (k) and the equation. Yes! k = 25/10 or 5/2 k = 10/4 or 5/2 Equation? y = 5/2 x  Is this a direct variation? If yes, give the constant of variation (k) and the equation. No! The k values are different! Which of the following is a direct variation? 1. A 2. B 3. C 4. D Which is the equation that describes the following table of values? 1. y = -2x 2. y = 2x 3. y= ½x 4. xy = 200 Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 28 when x=7, Find x when y = 52. HOW??? 2 step process 1. Find the constant variation k = y/x or k = 28/7 = 4 k=4 2. Use y = kx. Find the unknown (x). 52= 4x or 52/4 = x Therefore: x= 13 X =13 when Y=52 Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 3 when x=9, Find y when x = 40.5. HOW??? 2 step process 1. Find the constant variation. k = y/x or k = 3/9 = 1/3 K = 1/3 2. Use y = kx. Find the unknown (x). Therefore: y= (1/3)40.5 X =40.5 when y= 13.5 Y=13.5  Using Direct Variation to find unknowns (y = kx) Given that y varies directly with x, and y = 6 when x=-5, Find y when x = -8. HOW??? 2 step process 1. Find the constant variation. k = y/x or k = 6/-5 = -1.2 k = -1.2 2. Use y = kx. Find the unknown (x). k=y/x ; so… -1.2=y/-8 Therefore: Rewrite for solving for “y” y= -1.2(-8) X =-8 when Y=9.6 Using Direct Variation to solve word problems Problem: Step One: Find points in table A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? Step Two: Find the constant Step Three: Use the equation variation and equation: to find the unknown. 400 =36.25x k = y/x or k = 290/8 or 36.25 400 =36.25x y = 36.25 x 36.25 36.25 or x = 11.03 Using Direct Variation to solve word problems Problem: Step One: Find points in table. Julio wages vary directly as the number of hours that he works. If his wages for 5 hours are $29.75, how much will they be for 30 hours Step Three: Use the equation to find the unknown. y=kx Step Two: Find the constant y=5.95(30) or Y=178.50 variation. k = y/x or k = 29.75/5 = 5.95  Direct Variation and its graph y = mx + b, m = slope and b = y-intercept With direction variation the equation is y = kx Note: m = k or the constant and b = 0 therefore the graph will always go through… the ORIGIN!!!!! Tell if the following graph is a Direct Variation or not. - Do Together No No Tell if the following graph is a Direct Variation or not. - Do Independently No Yes NO No