Name:__________________________________________________ Date: ________________ Period: ______
Algebra 2
2-2 and 8-1 Day 1
Direct and Inverse Variation
Direct Variation: ___________________________________________________________________________
_________________________________________________________________________________________.
Constant of Variation: _______________________________________________________________________
_________________________________________________________________________________________.
Identifying Direct Variation from Tables
For each function, determine whether y varies directly with x. If so, what is the constant of
variation and the function rule?
Examples:
1.
2.
x
y
x
y
1
4
1
2
2
8
3
6
3
11
4
8
𝑦
=
𝑥
=
𝑦
=
𝑥
=
=
=
Try these:
3.
4.
x
3
2
-14
𝑦
=
𝑥
=
y
-21
-14
-7
=
x
2
3
6
𝑦
𝑥
=
y
5
7
13
=
=
Identifying Direct Variation from Equations
For each function, determine whether y varies directly with x. If so, what is the constant of
variation?
Examples:
1. 3𝑦 = 7𝑥
2. 7𝑦 = 14𝑥 + 7
Try these:
3. 5𝑥 + 3𝑦 = 0
4. 𝑦 =
𝑥
9
Using Proportions to Solve a Direct Variation
Suppose y varies directly with 𝑥, and 𝑦 = 9 when 𝑥 = −15. What is 𝑦 when 𝑥 = 21?
Try this:
1. Suppose y varies directly with 𝑥, and 𝑦 = 15 when 𝑥 = 3. What is 𝑦 when 𝑥 = 12?
Graphing Direct Variation Equations
What is the graph each direct variation equation?
𝑦=
3
𝑥
4
x
4
8
12
y
Try this:
2
1. 𝑦 = − 3 𝑥
x
y
Inverse Variation: ___________________________________________________________________________
_________________________________________________________________________________________.

When two quantities vary inversely, as one quantity __________________, the other _______________
proportionally.
Example:
1.
x
2
4
10
15
y
15
7.5
3
2


As 𝑥 _____________, 𝑦 ________________.
Test to see if 𝑥𝑦 is constant by __________________ 𝑥 and 𝑦.
Function Model: __________________
Try these:
Is the relationship between the variables a direct variation, an inverse variation, or neither? Write a function
model for the direct and inverse variations.
1.
2.
x
0.2
0.5
1.0
1.5
y
8
20
40
60
x
0.2
0.5
1.0
2.0
y
40
16
8.0
4.0
Determining an inverse variation
Suppose 𝑥 and 𝑦 vary inversely, and 𝑥 = 4 when 𝑦 = 12. What function models the inverse variation?
𝑘
𝑦=𝑥
1. Start with the general function form for inverse variation.
2. Substitute for 𝑥 and 𝑦.
3. Solve for 𝑘.
What is 𝑦 when 𝑥 = 10?
x
3
y
16
Combined Variation:_________________________________________________________________________
_______________________________________________________________________________________.
Joint Variation: ___________________________________________________________________________
_______________________________________________________________________________________.
Combined Variation
z varies jointly with x and y
Z varies jointly with x and y and inversely with w
Z varies directly with z and inversely with the product wy
Equation Form
𝑧 = 𝑘𝑥𝑦
𝑘𝑥𝑦
𝑤
𝑘𝑥
𝑧=
𝑤𝑦
𝑧=
Example:
The number of bags of grass seed n needed to reseed a yard varies directly with the area a to be seeded and
inversely with the weight w of a bag of seed. If it takes two 3-lb bags to seed an area of 36000 ft2, how many 3lb bags will seed 9000 ft2?
𝑛=
𝑘𝑎
𝑤
The combined variation equation is _____________________.