Lesson 3 • 35 min read
Variation
In This Lesson
1Direct Variation
y = kx
"y varies directly as x" or "y is directly proportional to x"
Characteristics
- • As x increases, y increases
- • As x decreases, y decreases
- • Ratio y/x is always constant (= k)
- • Graph passes through origin (0,0)
Real-World Examples
- • Distance = Speed × Time
- • Total cost = Price × Quantity
- • Circumference = π × Diameter
- • Wage = Hourly rate × Hours
Example: y varies directly as x. If y = 15 when x = 3, find y when x = 7.
Step 1: Find k using y = kx
15 = k(3) → k = 5
Step 2: Use k to find new y
y = 5(7) = 35
Direct Variation with Powers
y = kx² → "y varies directly as the square of x"
y = kx³ → "y varies directly as the cube of x"
2Inverse Variation
y = k/x or xy = k
"y varies inversely as x" or "y is inversely proportional to x"
Characteristics
- • As x increases, y DECREASES
- • As x decreases, y INCREASES
- • Product xy is always constant (= k)
- • Graph is a hyperbola
Real-World Examples
- • Speed = Distance / Time
- • More workers = Less time per task
- • Pressure and Volume (gases)
- • Price and Quantity (fixed budget)
Example: y varies inversely as x. If y = 6 when x = 4, find y when x = 8.
Step 1: Find k using xy = k
(4)(6) = k → k = 24
Step 2: Use k to find new y
(8)y = 24 → y = 3
Example 2: 5 workers can finish a job in 12 days. How long for 10 workers?
Workers × Days = k (constant work)
5 × 12 = k → k = 60
10 × d = 60 → d = 6 days
3Joint Variation
y = kxz
"y varies jointly as x and z"
Joint variation occurs when a variable depends on TWO or more other variables directly. It's like direct variation with multiple variables.
Real-World Examples
- • Area of rectangle: A = lw (Area varies jointly as length and width)
- • Simple interest: I = Prt (Interest varies jointly as principal, rate, and time)
- • Volume of cylinder: V = πr²h
Example: y varies jointly as x and z. If y = 60 when x = 3 and z = 4, find y when x = 5 and z = 2.
Step 1: Find k using y = kxz
60 = k(3)(4) → 60 = 12k → k = 5
Step 2: Use k to find new y
y = 5(5)(2) = 50
4Combined Variation
y = kx/z
"y varies directly as x and inversely as z"
Combined variation mixes both direct and inverse variation in one equation. Some variables are in the numerator (direct) and some in the denominator (inverse).
Common Forms
y = kx/z → y varies directly as x, inversely as z
y = kxz/w → y varies jointly as x and z, inversely as w
y = kx²/z → y varies directly as x², inversely as z
Example: y varies directly as x and inversely as z. If y = 10 when x = 4 and z = 2, find y when x = 6 and z = 3.
Step 1: Find k using y = kx/z
10 = k(4)/(2) → 10 = 2k → k = 5
Step 2: Use k to find new y
y = 5(6)/(3) = 30/3 = 10
Example 2: y varies jointly as x and z, and inversely as w. If y = 12 when x = 2, z = 3, w = 4, find y when x = 3, z = 5, w = 6.
y = kxz/w
12 = k(2)(3)/(4) → 12 = 6k/4 → 12 = 1.5k → k = 8
y = 8(3)(5)/(6) = 120/6 = 20
5Solving Variation Problems
Universal Steps
- Translate the statement into an equation with k
- Substitute the given values to find k
- Rewrite the equation with the value of k
- Substitute new values and solve
Translation Guide
"y varies directly as x"
y = kx
"y varies inversely as x"
y = k/x
"y varies jointly as x and z"
y = kxz
"y varies directly as x² and inversely as z"
y = kx²/z
Key Reminders
- • k is called the constant of variation or constant of proportionality
- • k is always positive in most real-world problems
- • "Varies as" means the same as "is proportional to"
- • Check your answer by plugging it back into the original equation