Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Direct Variation
A dripping faucet wastes one cup of water for every three minutes it drips.
1. Use the table to show how much water is wasted as time passes.
# of minutes
Cups of water
wasted
1
3
6
9
12
30
2. There is a relationship between the number cups of water wasted (w) and the
number of minutes (m) that passes. It can be expressed with a simple mathematical
operation. Using an equation, or in words, express this relationship.
Definition: Two variables are said to vary directly if their quotient is constant.
3. Explain, in words, why the dripping faucet example shows direct variation.
4. In the above situation, as time increases, the amount of water that has dripped
increases/decreases (circle the correct word).
5. Fill out the y-values for the following table:
𝑦
x
= 12
𝑥
In this table, as x values increases, the y values
1
increase/decrease.
2
3
𝑦
4
Is 𝑥 = 12 an example of direct variation? Why or why not?
6. Fill out the y-values for the following table:
x
xy=12
In this table, as x values increases, the y values
1
increase/decrease.
2
3
4
Is xy=12 an example of direct variation? Why or why not?
Name _____________________________________________ Date ___________
7. Fill out the y-values for the following table:
x
y=x+3
In this table, as x values increases, the y values
1
increase/decrease.
2
3
4
Is y = x + 3 an example of direct variation? Why or why
not?
Writing equations of Direct Variation given initial conditions.
Example: x and y vary directly. If x=12 when y=2, then what is x when y=24?
Solution: In this question, you are not being asked if variables vary directly, you’re being
TOLD they vary directly! Since they vary directly, that means that the quotient is
constant.
Set up an equation where you are showing that no matter what, the quotient is
constant. This means that for any pair of values x and y, when you divide the values, the
result will be the same.
Set up the equation:
𝑦1
𝑥1
=
𝑦2
𝑥2
Replace the appropriate x and y values:
Solve, by cross-multiplying.
Determining the Constant of Variation.
The constant of variation, in this case, is the constant that you get when you divide the
variables each time.
What is the constant of variation in the above problem?
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Direct Variation HOMEWORK
Determine if each of the following represent DIRECT variation. If your
answer is “yes,” determine the constant of variation.
a.
X
Y
1
10
4
9
7
8
b.
x
y
x
y
x
y
11
33
3
21
6
18
12
36
4
28
10
30
15
45
6
42
d. 14
42
c.
For each problem above, determine the constant of variation.
Algebra2/Trig Apps: Inverse Variation
Name _____________________________________________ Date ___________
A limo costs $800 to rent for the evening. The limo can fit a maximum of 10 couples. If
each couple shares the cost of the limo, what is the cost per couple?
1. Fill out the table of values to show the cost per couple.
# of couples
Cost per couple
1
$800
2
4
8
10
2. There is a relationship between the number of couples (n) and the cost per couple
(c). It can be expressed with a simple mathematical operation. Using an equation,
or in words, express this relationship.
Definition: Two variables are said to vary inversely if their product is constant.
3. In the above situation, as the number of couples increase, the cost per couple
increases/decreases (circle the correct word).
4. Fill out the y-values for the following table:
𝑦
x
= 24
𝑥
In this table, as x values increases, the y values
1
increase/decrease.
2
3
4
𝑦
Is 𝑥 = 24 an example of inverse variation? Why or why not?
5. Fill out the y-values for the following table:
x
xy=8
In this table, as x values increases, the y values
1
increase/decrease.
2
4
8
Is xy=8 an example of inverse variation? Why or why not?
Name _____________________________________________ Date ___________
6. Fill out the y-values for the following table:
x
y=5-x
In this table, as x values increases, the y values
1
increase/decrease.
2
3
4
Is y = 5 - x an example of direct variation? Why or why not?
Writing equations of Inverse Variation given initial conditions.
Example: x and y vary inversely. If x=12 when y=2, then what is x when y=24?
Solution: In this question, you are not being asked if variables vary inversely, you’re
being TOLD they vary inversely! Since they vary inversely, that means that the product is
constant.
Set up an equation where you are showing that no matter what, the quotient is
constant. This means that for any pair of values x and y, when you divide the values, the
result will be the same.
Set up the equation:
𝑥1 ∙ 𝑦1 = 𝑥2 ∙ 𝑦2
Replace the appropriate x and y values:
Solve.
Determining the Constant of Variation.
The constant of variation, in this case, is the constant that you get when you multiply the
variables each time.
What is the constant of variation in the above problem?
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Inverse Variation HOMEWORK
Determine if each of the following represent INVERSE variation. If your
answer is “yes,” determine the constant of variation.
1.
X
Y
x
y
x
y
X
Y
1
10
2
27
0.5
6
75
15
4
9
6
9
1
3
85
10
7
8
18
3
1.5
2
90
5
2.
3.
For each of the above, determine the constant of variation.
4.
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Direct versus Inverse Variation
Which type of variation is represented by each of the following: inverse, direct, or
neither? If you answer direct or inverse, write the equation.
1.
2.
3.
4.
5.
6.
7. Heart rates and life spans of most mammals are inversely related.
Mammal
Heart rate in beats per
Life span (in minutes)
minute
Mouse
634
1,567,800
Rabbit
158
6,307,200
Lion
76
13,140,000
Horse
63
15,768,000
a. Use the data to write a function that models this inverse variation.
b.
Use the equation you found in (a) to determine the approximate life span of a
squirrel if you are given that its heart rate is 190 beats per minute.
c. Use the equation you found in (a) to determine the approximate heart rate of an
elephant if you are given that its life span is about 70 years.
Name _____________________________________________ Date ___________
Algebra 2 Trig/Apps Homework
Suppose that x and y vary inversely. Write a function that models each inverse
variation.
1. x = 1 when y = 11
2. x = - 13 when y = 100
3. x = 1 when y = 1
4. x = 28 when y = -2
5. x = 1.2 when y = 3
6. x = 2.5 when y = 100
Is the relationship between the values in each table a direct variation, an inverse
variation, or neither? Write equations to model the direct and inverse variations.
Suppose that x and y vary inversely. Write a function that models each inverse
variation and find y when x = 10.
13. x = 20 when y = 5.
14. x = 20 when y = -4.
15. x = 5 when y = -
𝟏
𝟑
Name _____________________________________________ Date ___________
WARM –UP
Tell if the data has a direct variation relationship; if it is, give the constant of variation
and writ the equation.
x
9
12
y
3
4
15
5
Joint Variation
Joint variation is the same as direct variation with two or more quantities.
That is: Joint variation is a variation where a quantity varies directly as the product of
two or more other quantities.
y = kxz
or
𝒚
𝒙𝒛
=𝒌
Example 1
Suppose y varies jointly as x and z. Find y when x = 6 and z = 30 if y = 7 when z = 10 and
x = 3.
Solution:
Use a proportion that relates the values.
𝑦
(6)(30)
=
7
(3)(10)
Cross Multiply
(7)(6)(30) = y(3)(10)
1260 = 30y
42 = y
Name _____________________________________________ Date ___________
Your Turn
2. Assume a varies jointly with b and c. If b = 2 and c = 3, find the value of a. Given
that a = 12 when b =1 and c = 6.
3. z varies jointly with x and y. Find z when x = 4 and y = 9 if x = 2 when y = 3 and
z = 60.
4. Suppose c varies jointly with d and the square of g. c = 30 when d = 15 and g = 2.
Find d when c = 6 and g = 8.
5. Suppose d varies jointly with r and t, and d = 110 when r = 55 and t = 2.
Find r when d = 40 and t = 3.
Name _____________________________________________ Date ___________
ALGEBRA 2/TRIG APPS
Homework Joint Variation
Mrs. von Stein
1. If y varies jointly as x and z, and y =33 when x = 9 and z = 12, find y when x = 16 and
z = 22.
2. If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4,
find f when g= 3 and h = 6.
3. Wind resistance varies jointly as an object’s surface area and velocity. If an object
traveling at 40 miles per hour with a surface area of 25 square feet experiences a
wind resistance of 225 Newtons, how fast must a car with 40 square feet of surface
area travel in order to experience a wind resistance of 270 Newtons?
4. For a given interest rate, simple interest varies jointly as principal and time. If $2000
left in an account for 4 years earns interest of $320, how much interest would be
earned if you deposit $5000 for 7 years?
5. If a varies jointly as b and the square root of c, and a = 21 when b = 5 and c = 36, find
a when b = 9 and c = 225.
6. The volume of a pyramid varies jointly as its height and the area of its base. A
pyramid with a height of 12 feet and a base with area of 23 square feet has a
volume of 92 cubic feet. Find the volume of a pyramid with a height of 17 feet and a
base with an area of 27 square feet.
Name _____________________________________________ Date ___________
Name _____________________________________________ Date ___________
Name _____________________________________________ Date ___________
Algebra 2 Trig/Apps.
SWBAT: Solve combined variation problems
Warm – Up
1) The brightness of illumination, I, of an object varies inversely as the square of its distance, d,
from the source of illumination. If I = 18 luxes when d = 4m,
a. Find the value of k.
_______________________
b. Find I when d = 3m.
_______________________
2) If d varies jointly as r and t, and d = 110 when r = 55 and t = 2, find r when d = 40 and t = 3.
_______________________
COMBINED VARIATION
Example 1
y varies directly as x and inversely as z. If y = 5 when x = 3 and z = 4, find y when x = 6 and z = 8.
Example 2
y varies directly as x2 and inversely as z. If y = 12 when x = 2 and z = 7, find y when x = 3 and z = 9.
Name _____________________________________________ Date ___________
YOUR TURN
3. y varies directly as x and inversely as z. If y = 10 when x = 9 and z = 12, find y when x = 16 and
z = 10.
4. x varies directly as y3 and inversely as z . If x = 7 when y = 2 and z = 4, find x when y = 3 and
z = 9.
5. The number of girls varied directly as the number of boys and inversely as the number of teachers. When
there were 50 girls, there were 20 teachers and 10 boys. How many boys were there when there were 10
girls and 100 teachers?
GRAPHING
Use the coordinate graphs to graph each of the following:
6. xy = -20
7.
𝑦
𝑥
=2
Name _____________________________________________ Date ___________
Algebra 2 Trig/Apps. COMBINED VARIATION HOMEWORK Due 9/26/13
1. Physics. The force F of gravity on a rocket varies directly with its mass m and inversely with
the square of its distance d from Earth. Write a model for this combined variation.
2. Horses varied directly as goats and inversely as pigs squared. When the barnyard contained
5 horses there were 4 pigs and only 2 goats. How many goats went with 6 pigs and 10 horses?
3. a) y varies jointly as x and w and inversely as the square of z. Find the equation of variation
when y = 100, x = 2, w= 4, and z= 20.
b) Then solve for y when x = 1, w =5 and z = 4.
Graph each of the following:
4. xy = 16
5.
𝑦
𝑥
= −3
Name _____________________________________________ Date ___________
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Writing Equations of Variation
Sentence
Example:
The area of an object varies
directly with the square of its
radius,
1. The area of an object
varies jointly(directly)
with the base and the
height of the
rectangle.
2. The volume of an
object varies jointly
with the area of its
base and its height.
3. The volume of an
object varies jointly
with the square of its
radius and its height.
4. The height of an
object varies directly
with its volume and
inversely with the
square of its radius.
5. The volume of an
object varies jointly
with its length, width,
and height.
6. The length of an
object varies directly
with its volume, and
inversely with the
product of its width
and height.
Variables needed
A = area of the object
r = radius of the object
Equation
(use “k” for the
constant of
variation.)
𝑨
𝟐 = 𝒌
𝒓
so
A = kr2
Extra Credit: What
is the object and
what is the value of
the constant?
Circle
k=π
Name _____________________________________________ Date ___________
Name _____________________________________________ Date ___________
How to determine if a relationship is direct variation,
inverse variation, or neither
both variables are
INCreasing OR
both DECreasing
Read vars
either left to
right or top
to bottom
One var is
INCreasing and the
other is DECreasing
Possibly DIRECT
variation.
Check to see if
QUOTIENT of each x and
y is constant.
DIRECT VARIATION
YES
x/y = k or
y/x = k
NO
NEITHER
INVERSE VARIATION
Possibly INVERSE
variation.
YES
Check to see if PRODUCT
of each x and y is
constant.
NO
xy=k
NEITHER
Working with problems where it is GIVEN that a relationship
is either direct variation or inverse variation.
Problem tells you it is
DIRECT variation
DIVIDE y/x to determine
the constant of
variation (k)
equation is y/x=k
Problem tells you it is
INVERSE variation
MULTIPLY xy to
determine the constant
of variation (k)
equation is xy=k
Name _____________________________________________ Date ___________
Test Review
Part I: Am I inverse, direct, or neither?
a) For each of the following, indicate if the table of values shows inverse variation,
direct variation, or neither.
b) If your answer is “inverse” or “direct,” write the exact equation (there should be
no “k” in your equation.)
1.
2.
3.
4.
Part II: Translations
Translate the following sentences into equations using “k” to represent the constant of
variation. (Each is an actual application of physics!  )
5.
6.
Part III: More Translations
7. z varies jointly with x and y.
a.
Using a “k” to
represent the constant,
write the equation.
b.
If x = 2 when y = 4
and z = 64, determine the
value of k.
c.
Using part a and b,
determine the value of z
when x = 4 and y = 10.
Name _____________________________________________ Date ___________
9. p varies directly with the square of x and inversely with y and z.
a. Using a “k” to
represent the
constant, write the
equation.
b.
If x = -4 when y = 2 ,
z= 10, and p = 40,
determine the value of k.
c.
Using part a and b,
determine the value of p
when x = -2, y = 6, and z =
1.
Part IV: Graphs
10. Sketch the graph of xy= -4 on the axes below.
Does this equation represent direct or
inverse variation? How can you tell?
𝑦
11. Sketch the graph of 𝑥 = −4 on the axes below.
Does this equation represent direct or
inverse variation? How can you tell?
12. Identify whether the following graphs are inverse or direct variation, and then write
the equation.