Identifying,%20writing,%20and%20graphing%20inverse%20variation

1
Lesson 64

• Identifying, writing, and graphing inverse
  variation

2
Review direct variation

• y kx
• As one variable increases, the other increases at
  a constant rate, k

3
Inverse variation

• Inverse variation is a relationship between 2
  variables whose product is a constant.
• The equation xy k or y k/x, where k is a
  nonzero constant, defines an inverse variation
  between x and y
• As x increases, y decreases and as x decreases, y
  increases
• Look for the words "varies inversely" or "is
  inversely proportional to"

4
Example of inverse variation

• The relationship between the length and width of
  a rectangle with a constant area is an inverse
  variation because in order for the area to stay
  the same, if the length increases, the width must
  decrease.
• L x W Area

5
exploration

• 1) draw a rectangle that has a width of 1 unit
  and a length of 16 units on graph paper
• 2)draw different rectangles with the same area
  but different widths and lengths
• 3) make a table and complete it after drawing 5
  other rectangles with the same area
• 4) what happens to the length of each rectangle
  as the width increases?
• 5) what will the product of the width and length
  always be?
• 6)write an equation solved for y showing this
  relationship.

6

• In a direct variation, y is equal to the product
  of a constant k and x or y kx
• In an inverse variation, y is equal to the
  quotient of k and x or
• y k/x

7
Identifying an inverse variation

• Tell whether each relationship is an inverse
  variation
• y/6 x solve for y
• 6(y/6) (x)6
• y 6x this a direct variation
• xy 5
• y 5/x inverse variation

8
Direct or inverse variation?

• x 36/y
• y/12 x
• 4yx
• 3xy9

9
Product rule for inverse variation

• If (x1, y1) and (x2,y2) are solutions of an
  inverse variation, then x1y1 x2y2

10
Using the product rule

• If y varies inversely as x and y 3 when x 12,
  find x when y 9
• Use the product rule x1y1 x2y2
• (12)(3) x2(9)
• 36 9x2
• 4 x2

11
practice

• If y varies inversely as x and y 5 when x 12,
  find x when y 3

12
Graphing an inverse variation

• Write an inverse variation relating x and y when
  y 8 and x 3. Then graph it.
• Find k
• xy k
• (3)(8) 24
• k 24 so xy24 y 24/x
• Make a table of values and plot the points

13
practice

• Write an inverse variation relating x and y when
  x 2 and y 6. Then graph the relationship.

14
Investigation 7

• Comparing direct and inverse variation

15
compare

• Direct variation is a relationship between 2
  variables whose ratio is constant. The equation y
  kx, where k is a nonzero constant called the
  constant of variation, shows direct variation
  between variables x and y.
• Identify the constant of variation, given that y
  varies directly with x
• y is 10 when x 2
• y is 3 when x 6

16
Word problem

• Alex walks 3 miles per hour. If he walks at that
  rate for twice as long, he will travel twice as
  far. The ratio of the distance and time is always
  the same.
• Identify the constant of variation- rate, k ?
• Write an equation of direct variation that
  relates Alex's time to his distance traveled

17
Inverse variation

• In inverse variation, as x increases, y
  decreases. An inverse variation describes a
  relationship between 2 variables whose product is
  a constant. The equation xyk , where k is a
  nonzero constant, defines an inverse variation
  between x and y
• Identify the constant of variation, given that y
  varies inversely with x. then write the constant
  of variation.
• y is 1 when x is 3
• y is 4 , when x is 1/2

18
Inverse variation

• The equation of inverse variation can be written
  as y k/x or xy k

19
Word problem

• Alex lives 4 miles from school. If he walks at a
  slower rate than normal, it will take him longer
  to reach his destination. In other words, the
  more time he spends walking home, the slower he
  is actually walking. This represents an inverse
  variation.
• Identify the constant of variation.
• Write an equation of inverse variation that
  relates Alex's time to his rate of speed.

20
graphing

• A direct variation graph is linear
• An inverse variation graph is not linear and
  never intersects the x-axis

21
(No Transcript)