Fundamental Theorem of Variation
Explore the Fundamental Theorem of Variation using a small group setting of Algebra II
students. Students will explore how multiplying the independent variable by a constant affects
direct and inverse variations.
In this exploration, students will be placed into groups of three. Each group will explore a direct
variation and an inverse variation.
1. Consider the direct variation y = x3
a. Each person in your group should choose a different value for the independent variable.
Record it in the following chart. Fill in the additional x values. After finding the values for x,
find the corresponding dependent values and record them in the chart.
x
Original value =
Doubled =
Tripled =
Quadrupled =
y = x3
b. Discuss your results with other members of your group and answer these questions. Then
be ready to share will the class your answers.
What happens to y when x is doubled?
What happens to y when x is tripled?
What happens to y when x is multiplied by four?
What would happen to y when x is multiplied by any nonzero constant c?
2. Consider the inverse variation y = 1/x3
a. Each person in your group should use the same independent value as in problem #1. Record
it in the following chart. Fill in the additional x values. After finding the values for x, find the
corresponding dependent values and record them in the chart (hint: leave as a fraction).
x
Original value =
Doubled =
Tripled =
Quadrupled =
y =1/ x3
b. Discuss your results with other members of your group and answer these questions. Be
ready to share your answers with the class.
What happens to y when x is doubled?
What happens to y when x is tripled?
What happens to y when x is multiplied by four?
What would happen to y when x is multiplied by any nonzero constant c?
3. Using the information from your discussion, answer the following questions.
a.
If y = x2 and x is multiplied by 3, what is the result on y?
b.
If y = 1/x2 and x is multiplied by 3, what is the result on y?
Fundamental Theorem of Variation Teacher notes
Explore the Fundamental Theorem of Variation using a small group setting of Algebra II
students. Students will explore how multiplying the independent variable by a constant affects
direct and inverse variations.
In this exploration, students will be placed into groups of three. Each group will explore a direct
variation and an inverse variation.
1. Consider the direct variation y = x3
a. Each person in your group should choose a different value for the independent variable.
Record it in the following chart. Fill in the additional x values. After finding the values for x,
find the corresponding dependent values and record them in the chart.
x
Original value = 3
Doubled = 6
Tripled = 9
Quadrupled = 12
y = x3
27
216
729
1728
Possible answers:
b. Discuss your results with other members of your group and answer these questions. Then
be ready to share will the class your answers.
What happens to y when x is doubled? (216/27 = 8) y is 8 or 23 times as big
What happens to y when x is tripled? (729/27 = 27) y is 27 or 33 times as big
What happens to y when x is multiplied by four? (1728/27 = 64) y is 64 or 43 times as big
What would happen to y when x is multiplied by any nonzero constant c? y is c3 times as big
Note: All students will have the same answers to b, no matter the values they choose in a.
2. Consider the inverse variation y = 1/x3
a. Each person in your group should use the same independent value as in problem #1. Record
it in the following chart. Fill in the additional x values. After finding the values for x, find the
corresponding dependent values and record them in the chart (hint: leave as a fraction).
x
Original value =
Doubled =
Tripled =
Quadrupled =
y =1/ x3
1/27
1/216
1/729
1/1728
b. Discuss your results with other members of your group and answer these questions. Be
ready to share your answers with the class.
What happens to y when x is doubled? y is 1/8 or 1/23 times as big
What happens to y when x is tripled? y is 1/27 or 1/33 times as big
What happens to y when x is multiplied by four? y is 1/64 or 1/43 times as big
What would happen to y when x is multiplied by any nonzero constant c? y is 1/c3 times as big
Note: All students will have the same answers to b, no matter the values they choose in a.
3. Using the information from your discussion, answer the following questions.
a.
If y = x2 and x is multiplied by 3, what is the result on y? y is 32 or 9 times as big
b.
If y = 1/x2 and x is multiplied by 3, what is the result on y? y is 1/32 or 1/9 times as big
Note: From this point I show several examples that end with the Fundamental Theorem of
Variation.
If y varies directly as the nth power of x ( y = kxn) and x is multiplied by a nonzero constant c,
then y is multiplied by cn.
If y varies inversely as the nth power of x ( y = k / xn) and x is multiplied by a nonzero constant c,
then y is multiplied by 1 / cn , or y is divided by cn.