Rate of Change and Direct Variation

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Rate of Change and Direct Variation
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Rate of Change

• DefinitionThe 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).

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• 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

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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 rise m (y2-y1) run
  (x2-x1)

• m 4-2 2 1 6-2 4 2

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Problems

1. Find the rate of change of the line passing
   through P (4,1) and Q (-5,-5).

1. Find the rate of change of the line passing
   through R (8,6) and S (5,10).

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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?

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Constant of Variation

• DefinitionThe 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

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Direct Variation

• DefinitionDirect 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

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Example
9 8 7 6 5 4 3 2 1
y

• If we are told that y8 and x4 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 dont 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

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y
9 8 7 6 5 4 3 2 1

• If we know that y varies directly as x varies,
  when y8 and x4, we can find x when y2 and when
  y-4. How?

• Use the formulasy kx or k y x

• Given the point (4,8) we just found
• k y 8 2 x 4

• When y 2, x ?
• y2 x? k2

• When y -4, x ?
• y-4 x? k2

x
0 1 2 3 4 5 6 7
8
y kx 2 2x x1 2 2
y kx -4 2x x-2 2 2
Our point on the graph (1,2)
Our point on the graph (-2,-4)
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Problems
If y varies directly with x, find the constant of
variation, k, and write an equation of direct
variation.

1. y - 40 when x 16

y kx or k y/x - 40
16k k - 40/16 16 16
k - 2.5- 2.5 k

1. y 3 when x 15

y kx or k y/x 3
15k k 3/1515 15
k 1/51/5 k