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Rate of Change and Direct Variation

Rate of Change and Direct Variation. Rate of Change. Definition: The rate of change refers to the steepness or slope that a line or a given object may have in reference to a starting point , (x 1 ,y 1 ), and any given stopping point , (x 2 ,y 2 ).

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Rate of Change and Direct Variation

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  1. Rate of Change and Direct Variation

  2. Rate of Change • Definition:The rate of change refers to the steepness or slope that a line or a given object may have in reference to a starting point, (x1,y1), and any given stopping point, (x2,y2).

  3. Slope (denoted by m) is the ratio of the rise of the line or object to the run of the line or object. • Written as a fraction, the ratio is: m = rise run • The rise is the difference of the vertical coordinates: y1 and y2. • The run is the difference of the horizontal coordinates: x1 and x2. • If rise=(y2-y1) and run=(x2-x1) then the ratio of the slope as a fraction in :m = y2-y1 x2-x1

  4. Example: y 8 7 6 5 4 3 2 1 • Given point P= (2,2) and point Q= (6,4) find the slope (rate of change) of the line passing through these points. Q P • Slope = rise run • Rise = (y2-y1) • Run = (x2-x1) x 0 1 2 3 4 5 6 7 8 • P = (2,2) Q = (6,4) (x1,y1) (x2,y2) • m = risem = (y2-y1) run (x2-x1) • m = 4-2 = 2 = 1 6-2 4 2

  5. P = (4,1) = (x1,y1) Q = (-5,-5) = (x2,y2) R = (8,6) = (x1,y1) S = (5,10) = (x2,y2) The slope or rate of change of the line passing through P & Q is 2 3 The slope or rate of change of the line passing through R & S is - 4 3 m = (y2 - y1) = (-5 - 1) = -6 = 2 (x2 - x1) (-5 - 4) -9 3 m = (y2 - y1) = (10 - 6) = - 4 (x2 - x1) (5 - 8) 3 Problems: • Find the rate of change of the line passing through P (4,1) and Q (-5,-5). • Find the rate of change of the line passing through R (8,6) and S (5,10).

  6. What if… • your rate of change is 0 ? 2 It is zero. • your rate of change is 7 ? 0 It is undefined. • Is this the same for slope?

  7. Constant of Variation • Definition:The constant of variation is the change in steepness from one point to the next. The constant of variation does not change as you move from one point to the next.This constant of variation is called k, where k can not be zero (k≠0). • Examples involving constant of variation: • Driving: speed and distance • Working: hourly pay and the amount you get paid • Painting: amount of paint needed to paint a room and the size of the room

  8. Direct Variation • Definition:Direct Variation means as one item changes the related item must change the same amount as k (just as the slope does). • So, if y varies directly as x varies, then y = kx or if we solve this literal equation for k we have: k = y x

  9. Example: 9 8 7 6 5 4 3 2 1 y • If we are told that y=8 and x=4 and we know that y varies directly as x varies, we can write an equation of direct variation. • This would be k= y xsince we don’t know k. x 0 1 2 3 4 5 6 7 8 • k = y = 8 = 2 x 4 • The equation using the given point would be: y = 2x

  10. y 9 8 7 6 5 4 3 2 1 • If we know that y varies directly as x varies, when y=8 and x=4, we can find x when y=2 and when y=-4. How? • Use the formulas:y = kx or k = y x • Given the point (4,8) we just found: k = y = 8 = 2 x 4 • When y = 2, x = ? • y=2 ; x=? ; k=2 • When y = -4, x = ? • y=-4; x=? ; k=2 x 0 1 2 3 4 5 6 7 8 y = kx 2 = 2x x=1 2 2 y = kx -4 = 2x x=-2 2 2 Our point on the graph: (1,2) Our point on the graph: (-2,-4)

  11. Problems: If y varies directly with x, find the constant of variation, k, and write an equation of direct variation. • y = - 40; when x = 16 y = kx or k = y/x - 40 = 16k k = - 40/16 16 16 k = - 2.5- 2.5 = k • y = 3; when x = 15 y = kx or k = y/x 3 = 15k k = 3/1515 15 k = 1/51/5 = k

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