Chapter 5A Rational Functions
5.1A Variation Functions
Objectives:
A.CED.2: Create equations in two or more variables to represent relationships between
quantities; graph equations on coordinate axes with labels and scales.
A.CED.3: Represent constraints by equations or inequalities, and by systems of equations
and/or inequalities, and interpret solutions as viable or nonviable options in a
modeling context.
For the Board: You will be able to solve problems involving direct, inverse, joint, and combined
variations.
Bell Work 5.1:
Solve each equation.
2.4 2
1.

x
9
2. 1.6x = 1.8(24.8)
Anticipatory Set:
There are four types of special functions called variation problems.
1. Direct variation
2. Inverse variation
3. Joint variation
4. Combined Variation
A direct variation is a relationship between two variables x and y such that y = kx, where k ≠ 0.
k is the constant of variation. We say y varies directly as x.
Open the book to page 313 and read example 1.
Example: If y varies directly as x, and y = 27 when x = 6, find y if x = 4.
Step 1: Write the formula.
y = kx
2: Replace the x and y with the given values.
3: Solve for k.
k = 9/2 or 4.5
4: Rewrite the formula.
y = 4.5x
5: Use the formula to solve the problem.
27 = 6k
y = 4.5(4) = 18
White Board Activity:
Practice: If y varies directly as x, and y = 6.5 when x = 13, find y if x = 20.
y = kx
6.5 = 13x
x = 0.5
y = 0.5x
y = 0.5(20) = 10
Open the book to page 314 and read example 2.
Example: The cost of an item in euros E varies directly as the cost of the item in dollars D.
E = 3.85 euros when D = $5. Find D when E = 10 euros.
e = kd
3.85 = k ∙ 5
k = 3.85/5 = 0.77
e = 0.77d
10 = 0.77d
d = 10/0.77 = $12.99
White Board Activity:
Practice: The perimeter P of a regular dodecagon varies directly as the side length S, and P = 18 in when
S = 1.5 in. Find S when P = 75 in.
P = ks
18 = k ∙ 1.5
k = 18/1.5 = 122
P = 12s
75 = 12s
s = 75/12 = 6.25 in.
An inverse variation is a relationship between two variables x and y such that y = k/x, where k ≠ 0.
We say y varies inversely as x.
Open the book to page 315 and read example 4.
Example: If y varies inversely as x, and y = 4 when x = 5, find y when x = 10.
y = k/x
4 = k/5
k = 20
y = 20/x
y = 20/10 = 2
White Board Activity:
Practice: If y varies inversely as x, and y = 4 when x = 10, find y when x = 30.
y = k/x
4 = k/10
k = 40
y = 40/30 = 4/3
Open the book to page 315 and read example 5.
Example: The time T needed to complete a race varies inversely as the runner’s average speed S.
If a runner with a speed of 8.82 mph completes the race in 2.97 hours, what is the
speed of a runner who completes the race in 3.5 hours?
T = k/S
2.97 = k/8.82
k = 26.1954
T = 26.1954/S
3.5 = 26.1954/S
S = 26.1954/3.5 = 7.48 mph
White Board Activity:
Practice: 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 hours, how many
hours would it take 15 volunteers to build a house?
T = k/V
62.5 = k/20 k = 125
T = 1250/V T = 125/15 = 83 1/3 h
A joint variation is a type of direct variation, where one quantity varies directly with two other quantities
using the formula y = kxz, where k ≠ 0. We say y varies jointly as x and z.
Open the book to page 314 and read example 3.
Example: If y varies jointly with x and z, and y = 20 when x = 2 and z = 4, find y when x = 5 and z = 7.
y = kxz
20 = k(2)(4) = 8k
k = 2.5
y = 2.5xz
y = 2.5(5)(7) = 87.5
White Board Activity:
Practice: If y varies jointly with x and z, and y = 36 when x = 2 and z = 3, find y when x = 9 and z = 4.
y = kxz
36 = k(2)(3)
k=6
y = 6xz
y = 6(9)(4) = 216
Example: The volume V of a pyramid varies jointly as the area of the base B and the height h.
V = 12 ft3 when B = 9 ft2 and h = 4 ft. Find B when V = 24 ft3 and h = 9 ft.
V = kBh
12 = k ∙ 9 ∙ 4 = 36k
k = 12/36 = 1/3
V = 1/3 Bh
2
24 = 1/3 ∙ B ∙ 9 = 3B B = 8 ft
White Board Activity:
Practice: The lateral surface area L of a pyramid varies jointly as the base perimeter P and the slant
height l. L = 1260 m2 when P = 35 m and l = 18 m. Find P when L = 8 m2 and l = 5 m.
L = kPl
1260 = k(35)(18)
1260 = 63k
k=½
L = ½ Pl
8 = ½ 5P= 2.5P
P = 8/2.5 = 3.2 m
A combined variation is a relationship that contains both direct and inverse variation.
Open the book to page 316 and read example 7.
Example: If y varies directly as x and inversely as z, and y = 50 when x = 2 and z = 5, find y when x = 4
and z = 8.
y = kx/z
50 = 2k/5 0.4k
k = 125
y = 125x/z
y = 125(4)/5 = 100
White Board Activity:
Practice: If y varies directly as x and inversely as z, and y = 10 when x = 5 and z = 2, find y when x = 4
and z = 10.
y = kx/z
10 = 5k/2 = 2.5k
k=4
y = 4x/z
y = 4(4)/10 = 1.6
Example: The change in temperature T of a wire varies inversely as its mass m and directly as the
heat energy E transferred. The temperature of a wire with a mass of 0.1 kg rises
5°C when 450 joules of heat energy are transferred to it. How much heat energy must be
transferred to a wire with a mass of 0.2 kg to raise its temperature 20°C?
T = kE/m
5 = k ∙ 450/0.1
k = 5 ∙ 0.1/450 = 0.001
T = 0.0011E/m
20 = 0.0011E/0.2 = 2600 jouls
White Board Activity:
Practice: The volume V of gas varies inversely as the pressure P and directly as the temperature T.
A certain gas has a volume of 10 liters, a temperature of 300 kelvins, and a pressure of
1.5 atmospheres. If the gas is heated to 400 kelvins and has a pressure of 1 atmosphere,
what is its volume?
V = kT/P
10 = k ∙ 300/1.5
k = 0.05
V = 0.05T/P
V = 0.05(400)/1 = 20 liters
Assessment:
Question student pairs.
Independent Practice:
Text: pgs. 317 – 318 prob. 2 – 12, 16 – 27, 31.
For a Grade:
Text: pgs. 317 – 318 prob. 12, 16, 20, 22.