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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 + b?) Another way of writing this is k =  In other words: *  As X increases in value, Y increases or * As X decreases in value, Y decreases.
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 16/8=k or k =2 Note k stays constant. y = 2x is the equation!
Note: X decreases, 30, 15, 9 And Y decreases. 10, 5, 3 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 9/3=k or k =3 Note k stays constant.  y = 3x is the 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 -10/-40=k or k = ¼ Note k stays constant.  y = ¼ x is the equation! Examples of Direct Variation:
What is the constant of variation for the following direct variation?  ,[object Object],[object Object],[object Object],[object Object],32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Answer Now
Is this a direct variation?  If yes, give the constant of variation (k) and the equation. Yes! k = 6/4 or 3/2 Equation? y = 3/2 x
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! Is this a direct variation?  If yes, give the constant of variation (k) and the equation.
Which of the following is a direct variation? ,[object Object],[object Object],[object Object],[object Object],32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Answer Now
Which is the equation that describes the following table of values?  ,[object Object],[object Object],[object Object],[object Object],32 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Answer Now
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  x= 13 Therefore: X =13 when Y=52
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). y= (1/3)40.5  y= 13.5 Therefore: X =40.5 when 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). y= -1.2(-8)  x= 9.6 Therefore: X =-8 when Y=9.6 Using Direct Variation to find unknowns (y = kx)
Using Direct Variation to solve word problems Problem: A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? Step One : Find points in table Step Two:   Find the constant variation and equation: k = y/x or k = 290/8 or 36.25 y = 36.25 x Step Three:   Use the equation  to find the unknown. 400  =36.25x  400   = 36.25x 36.25 36.25 or  x = 11.03
Using Direct Variation to solve word problems Problem: 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 One:  Find points in table. Step Two:   Find the constant variation. k = y/x or k = 29.75/5 = 5.95 Step Three:   Use the equation  to find the unknown. y=kx y=5.95(30) or  Y=178.50
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. No Yes No No
No Yes Yes No Tell if the following graph is a Direct Variation or not.

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5.2 Directvariation

  • 1. Direct Variation What is it and how do I know when I see it?
  • 2. Definition: Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx + b?) Another way of writing this is k = In other words: * As X increases in value, Y increases or * As X decreases in value, Y decreases.
  • 3. 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 16/8=k or k =2 Note k stays constant. y = 2x is the equation!
  • 4. Note: X decreases, 30, 15, 9 And Y decreases. 10, 5, 3 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 9/3=k or k =3 Note k stays constant. y = 3x is the equation! Examples of Direct Variation:
  • 5. 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 -10/-40=k or k = ¼ Note k stays constant. y = ¼ x is the equation! Examples of Direct Variation:
  • 6.
  • 7. Is this a direct variation? If yes, give the constant of variation (k) and the equation. Yes! k = 6/4 or 3/2 Equation? y = 3/2 x
  • 8. 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.
  • 9. No! The k values are different! Is this a direct variation? If yes, give the constant of variation (k) and the equation.
  • 10.
  • 11.
  • 12. 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 x= 13 Therefore: X =13 when Y=52
  • 13. 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). y= (1/3)40.5 y= 13.5 Therefore: X =40.5 when Y=13.5 Using Direct Variation to find unknowns (y = kx)
  • 14. 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). y= -1.2(-8) x= 9.6 Therefore: X =-8 when Y=9.6 Using Direct Variation to find unknowns (y = kx)
  • 15. Using Direct Variation to solve word problems Problem: A car uses 8 gallons of gasoline to travel 290 miles. How much gasoline will the car use to travel 400 miles? Step One : Find points in table Step Two: Find the constant variation and equation: k = y/x or k = 290/8 or 36.25 y = 36.25 x Step Three: Use the equation to find the unknown. 400 =36.25x 400 = 36.25x 36.25 36.25 or x = 11.03
  • 16. Using Direct Variation to solve word problems Problem: 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 One: Find points in table. Step Two: Find the constant variation. k = y/x or k = 29.75/5 = 5.95 Step Three: Use the equation to find the unknown. y=kx y=5.95(30) or Y=178.50
  • 17. 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…
  • 19. Tell if the following graph is a Direct Variation or not. No Yes No No
  • 20. No Yes Yes No Tell if the following graph is a Direct Variation or not.