7.11 Variation Functions

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7.11 Variation Functions

• Mrs. Cassidy it is so weird! The more I study,
  the worse I do!!

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• Babysitting, my niece earns 12 for
• 2 hours, 18 for 3 hours.
• 2 cups 1 pint and 8 cups 4 pints
• When I walked to high school at a rate of 4 miles
  per hour, it took me half an hour. When
• I was running late and walked at
• a rate of 5 miles per hour, it
• took me 24 minutes.

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Variation Functions
Direct
Inverse
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Variation Functions
Direct
Inverse
Constant ? Variable
Constant ?Variable
Lets see what this looks like
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Direct Variation Functions

• n gt 0
• These are called Polynomial functions
• What did you notice with the graph?
• As x went up

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Inverse Variation Functions

• n lt 0
• These are called Polynomial functions
• What did you notice with the graph?
• As x goes up
• The x and y axis will act as asymptotes

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What if no one tells you

• How do you tell from a graph whether a function
  is a direct variation or inverse variation?
• How do you tell from a table of values whether a
  function is direct variation or inverse variation?

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Lets try some

• Babysitting, my niece earns 12 for 2 hours, 18
  for 3 hours.
• If x hours and y what she earns, what could
  be an equation?
• When kickboxing last week, I noticed that it took
  5 lbs of pressure to break a board that was 2
  feet long but only 1 2/3 lbs to break a board
  that is 6 feet long.
• If x length and y pounds of pressure, what
  could be an equation?

y 6x
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What if it isnt quite so obvious?

• There are 3 steps to these types of problems
• 1. Determine whether it is a direct variation or
  inverse variation problem.
• 2. Look for patterns (well practice this!)
• 3. Predict requested value

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Add-Add Property of Linear Functions

• Given f(x) 7x, find f(2), f(5) and f(8)
• For linear functions, adding a constant to x adds
  the constant (not necessarily the same one) to y.
  

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Add- Multiply Property of Exponential Functions

• Given the function f(x) 2(3x), find f(1), f(3)
  and f(5)
• For exponential functions, adding a constant to x
  multiplies y by a constant.

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Multiply-Multiply Property of Variation Functions

• Given the function f(x) 5x2, find f(1), f(2),
  f(4) and f(8)
• What does it look like you are actually
  multiplying each term by?

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Multiply-Multiply Property of Variation Functions

• If y kxn, then multiplying the value of x by
  the constant c multiplies the value of y by the
  constant cn.

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Multiply-Multiply Property of Variation Functions

• If y kxn, then multiplying the value of x by
  the constant c multiplies the value of y by the
  constant cn.
• If , then multiplying the value of
  x by the
• constant c divides the value of y by the
  constant cn.

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Practice Phrases

• y varies directly with x
• y varies linearly with x
• y varies inversely with x
• y is inversely proportional to x
• y is directly proportional to the cube of x
• y decreases exponentially with x
• y increases exponentially with x
• y varies inversely with the square of x
• y is a quadratic function of x
• y is a constant function

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Lets Practice

• Example 1
• In a lightning storm, the time interval between
  the flash and bang is directly proportional to
  the distance between you and the lightning.
• A) Variables? Which should be independent?
• b bang d distance distance is
  independent
• B) If the thunder clap from lightening 5 km away
  takes 15 seconds to reach you, write the
  particular equation.

b5d
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Lets Practice (cont)

• C) What is the label on the constant?
• s/km
• D) Calculate the times for the thunder sound to
  reach you from lightning bolts which are 1, 2.5
  and 10 km away.
• 3, 7.5 and 30 seconds

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Lets Practice more ?

• Example 2
• The intensity of radiation received for tumor
  treatment depends on the distance from a source.
  Suppose for that particular source, the intensity
  is 80 mr/hr at 2 meters and 5 mr/hr at 8 meters.
• A) What is the general equation? What is the
  particular equation?

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Lets Practice more ?

• Example 2
• The intensity of radiation received for tumor
  treatment depends on the distance from a source.
  Suppose for that particular source, the intensity
  is 80 mr/hr at 2 meters and 5 mr/hr at 8 meters.
• A) What is the general equation? What is the
  particular equation?

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Lets Practice more ? (cont)

• C) What would the intensity be if your distance
  were 16 meters? 10 meters? 10 centimeters?
• D) At what distance would the intensity be 0.5
  meters?
• 25 meters

1.25
3.2
32,000
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More? OF course!

• Truth in Advertising In a claim in an old Time
  Magazine advertisement for South African Airways,
  one Ostrich egg is equivalent to two dozen
  chicken eggs.

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OK.

• In your group, come up with the function (make up
  anything you want, within reason ?) and find two
  x/y values.
• On another piece of paper, write the two sets of
  x/y values down, and then give them to another
  group.

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Resources

• http//mathforum.org/library/drmath/view/57504.htm
  l
• http//www.regentsprep.org/Regents/math/algtrig/AT
  E7/Inverse20Variation.htm
• http//quiz.uprm.edu/tutorials/direct_var/direct_v
  ar_right.xml
• Your textbook