Transcript 7.11 Variation Functions

7.11 Variation Functions
Mrs. Cassidy – it is so weird! The
more I study, the worse I do!!
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.
Variation Functions
Direct
Inverse
Variation Functions
Direct
Inverse
y  kx
k
y n
x
n
Constant Variable
Lets see what this looks like…
Constant  Variable
Direct Variation Functions
y  kx
n
n>0
These are called Polynomial functions
What did you notice with the graph?
As x went up…
Inverse Variation Functions
k
y n
x
n<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
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?
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?
y = 6x
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?
10
y
x
What if it isn’t 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 (we’ll practice this!)
3. Predict requested value
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.
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.
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?
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.
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.
k
If y  n , then multiplying the value of x by
the x
constant c divides the value of y by the constant
cn.
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
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.
b=5d
Let’s 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
Let’s 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?
Let’s 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?
k
320
R 2
R= 2
d
d
Let’s Practice more  (cont)
C) What would the intensity be if your distance
were 16 meters? 10 meters? 10 centimeters?
1.25
3.2
32,000
D) At what distance would the intensity be 0.5
meters?
25 meters
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.
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.
Resources
http://mathforum.org/library/drmath/view/57504.html
http://www.regentsprep.org/Regents/math/algtrig/ATE7/Inverse%20Variation.htm
http://quiz.uprm.edu/tutorials/direct_var/direct_var_right.xml
Your textbook