Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
Definition: Two variables are said to vary directly if their quotient is constant.
𝒚
𝒙
=𝒌
Definition: Two variables are said to vary inversely if their product is constant. xy = k
Definition: Joint Variation, where more than two variables are related directly
y = kxz
or
𝒚
𝒙𝒛
=𝒌
Definition: Combined Variation, which involves a combination of direct or joint variation, and inverse
variation
Examples:





Direct: The number of dollars I make varies directly (or you can say varies proportionally) with
how much I work.
Direct: The length of the side a square varies directly with the perimeter of the square.
Inverse: The number of people I invite to my bowling party varies inversely with the number of
games they might get to play (or you can say is proportional to the inverse of).
Inverse: The temperature in my house varies indirectly (same as inversely) with the amount of
time the air conditioning is running.
Inverse: My GPA may vary inversely with the number of hours I watch TV
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Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
How to determine if a relationship is direct variation,
inverse variation, or neither
Possibly DIRECT
variation.
both variables are
INCreasing OR
both DECreasing
Check to see if
QUOTIENT of each x and
y is constant.
Read vars
either left to
right or top
to bottom
One var is
INCreasing and the
other is DECreasing
YES
DIRECT VARIATION
𝑦
=k
𝑥
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
2
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
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.
Which type of variation is represented by each of the following: inverse, direct, or
neither? If your answer is not “neither,” what equation relates x and y?
5.
6.
7.
8.
9.
10.
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Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
Part II: Translations
Translate the following sentences into equations using “k” to represent the constant
of variation. (Each is an actual application of physics!  )
11. Suppose you spin a circular weight on the end of a string. The force F on the
string varies directly with the square of the velocity v of the weight.
12. Suppose you spin a circular weight on the end of a string. The force F varies
inversely with the radius r of the circle.
Part III: More Translations
13. 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.
14. 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.
4
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
15. Suppose y varies jointly with x and the square root of z.
a. Using a “k” to represent
the constant, write the
equation.
b. If y = -18 when x = 2,
and z = 9, determine
the value of k.
c. Using parts a and b,
find x when y = 10 and
z = 4.
16. If y varies jointly as x and w and inversely as the square of z.
a. Using a “k” to represent
the constant, write the
equation.
b. If y = 100, x = 2, w = 4,
and z = 20 determine
the value of k.
c. Find y when x = 1, w = 5,
and z = 4.
Part IV: Graphs
17. Sketch the graph of xy= -4 on the axes below.
Does this equation represent direct or
inverse variation? How can you tell?
5
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
18. a) Sketch the graph of
𝑦
𝑥
= −4 on the axes below.
b) Does this equation represent direct or inverse variation? _______________
c) How can you tell? ____________________________________________________
19. a) Identify whether the following graphs are inverse or direct variation and then
b) write the equation.
a)_______________________
a) _________________________
b) ______________________
b) __________________________
6
Name _____________________________________________ Date ___________
Algebra2/Trig Apps: Review Topic 1: Variation
Part V Writing Translations
Write a variation statement for each of the following models in which k is the
constant of variation.
20. V 
k
___________________________________________________________________
t2
21. R  klwh _________________________________________________________________
22. G 
kp1 p2
________________________________________________________________
d
7