Chapter 5-1 Variation Functions
Obj: To solve problems involving direct, inverse, joint, and combined variation.
(WHY? – Variation functions can be used to determine how many people are needed to complete a task, such
as building a home, in a given time – See example 5)
Direct Variation is a relationship between two variables x & y that can be
written in the form: 𝑦 = 𝑘𝑥, 𝑤ℎ𝑒𝑟𝑒 𝑘 ≠ 0, 𝑎𝑛𝑑 𝑘 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (also, it’s a
linear equation 𝑦 = 𝑚𝑥 + 𝑏 𝑤ℎ𝑒𝑟𝑒 𝑏 = 0.)
Solving Direct Variation Problems
Ex 1 (p.313) Given: y varies directly as x, and y = 14 when x = 3.5. Write
the direct variation function. (hint: find k)
You try…y varies directly as x, and y = 6.5 when x = 13, write the direct
variation function.
When you want to find specific values in a direct variation problem, you
can solve for k and then use substitution or use the proportion derived
below.
𝑦1 = 𝑘𝑥1 →
𝑦1
𝑥1
= 𝑘 𝒂𝒏𝒅 𝑦2 = 𝑘𝑥2 →
𝑦2
𝑥2
= 𝑘 𝒔𝒐
𝑦1
𝑥1
=
𝑦2
𝑥2
Ex 2 (p.314) – The circumference of a circle 𝐶 varies directly as the radius
𝑟, and 𝐶 = 7𝜋 𝑓𝑡 𝑤ℎ𝑒𝑛 𝑟 = 3.5 𝑓𝑡. 𝐹𝑖𝑛𝑑 𝑟 𝑤ℎ𝑒𝑛 𝐶 = 4.5𝜋 𝑓𝑡.
Joint Variation is a relationship among three variables that can be written
in the form:
𝑦 = 𝑘𝑥𝑧, 𝑤ℎ𝑒𝑟𝑒 𝑘 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. 𝐹𝑜𝑟 𝑦 = 𝑘𝑥𝑧,
𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑗𝑜𝑖𝑛𝑡𝑙𝑦 𝑎𝑠 𝑥 𝑎𝑛𝑑 𝑧
Solving Joint Variation Problems
Ex 3 (p. 314) The area 𝐴 of a triangle varies jointly as the base 𝑏, and the
height ℎ, 𝐴 = 12 𝑚2 𝑤ℎ𝑒𝑛 𝑏 = 6 𝑚 𝑎𝑛𝑑 ℎ = 4𝑚.
Find 𝑏 𝑤ℎ𝑒𝑛 𝐴 = 36𝑚2 𝑎𝑛𝑑 ℎ = 8 𝑚
Step 1: Find k
Step 2: Use the variation function
You try…The lateral surface area 𝐿 of a cone varies jointly as the base
radius 𝑟 and the slant height 𝑙, and 𝐿 = 63𝜋 𝑚2 𝑤ℎ𝑒𝑛 𝑟 = 3.5 𝑚 𝑎𝑛𝑑
𝑙 = 18 𝑚. Find 𝑟 to the nearest tenth when 𝐿 = 8𝜋 𝑚2 𝑎𝑛𝑑 𝑙 = 5 𝑚
Inverse Variation is a relationship between two variables, x and y, that can
𝑘
be written in the form:
𝑦 = , 𝑤ℎ𝑒𝑟𝑒 𝑘 ≠
𝑥
𝑘
0. 𝐹𝑜𝑟 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑦 = , 𝑦 𝑣𝑎𝑟𝑖𝑒𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑙𝑦 𝑎𝑠 𝑥
𝑥
Inverse Variation Problems
Ex 4 (p.315) Given: y varies inversely as x, and y = 3 when x = 8. Write
the inverse variation function. (again…find k!)
You try… Given: y varies inversely as x, and y = 4 when x = 10. Write the
inverse variation function.
When you want to find specific values in an inverse variation problem, you
can solve for k and then use substitution or use the equation derived
below.
𝑦1 =
𝑘
𝑘
→ 𝑦1 𝑥1 = 𝑘 𝒂𝒏𝒅 𝑦2 =
→ 𝑦2 𝑥2 = 𝑘 𝒔𝒐 𝑦1 𝑥1 = 𝑦2 𝑥2
𝑥1
𝑥2
Ex 5 (p.315) The time t that it takes for a group of volunteers to construct
a house varies inversely as the number of volunteers v. If 20 volunteers
can build a house in 62.5 working hoours, how many volunteers would be
needed to build a house in 50 working hours?
REMEMBER…….
Homework Ch 5-1/ 5.1 – Pg 317, 1 – 12 (do not graph)