Direct and Inverse Variation

Direct Variation

Direct variation: Two variables are related by multiplication

Formula: y = kx

Where:

• k = constant of variation (always the same)
• As x increases, y increases proportionally
• As x decreases, y decreases proportionally

In words: "y varies directly with x"

Identifying Direct Variation

Characteristics:

• Equation is y = kx (or can be written that way)
• Ratio y/x is constant
• Graph is a line through the origin
• y/x = k for all points

Example 1: Is It Direct Variation?

Given: (2, 6), (4, 12), (5, 15)

Check ratios:

• 6/2 = 3
• 12/4 = 3
• 15/5 = 3

All equal to 3! → Yes, direct variation with k = 3

Equation: y = 3x

Example 2: Not Direct Variation

Given: (1, 5), (2, 8), (3, 11)

Check ratios:

• 5/1 = 5
• 8/2 = 4
• 11/3 ≈ 3.67

Ratios not constant → Not direct variation

Note: This is linear (y = 3x + 2) but NOT direct variation

Finding the Constant k

Given one pair of values, find k:

k = y/x

Example: Find k and Equation

y varies directly with x. When x = 4, y = 20.

Find k:

k = y/x = 20/4 = 5

Equation: y = 5x

Find y when x = 7:

y = 5(7) = 35

Solving Direct Variation Problems

Example: Distance and Time

Distance varies directly with time at constant speed.

If you travel 120 miles in 2 hours, how far in 5 hours?

Step 1: Find k (speed)

• 120 = k(2)
• k = 60 mph

Step 2: Use equation d = 60t

• d = 60(5) = 300 miles

Answer: 300 miles

Example: Pay and Hours

Pay varies directly with hours worked.

Earn $84 for 6 hours. Find pay for 10 hours.

Find k:

• 84 = k(6)
• k = 14 (dollars per hour)

Find pay:

• y = 14(10) = $140

Answer: $140

Inverse Variation

Inverse variation: As one variable increases, the other decreases

Formula: y = k/x or xy = k

Where:

• k = constant of variation
• As x increases, y decreases
• As x decreases, y increases

In words: "y varies inversely with x"

Identifying Inverse Variation

Characteristics:

• Equation is y = k/x (or xy = k)
• Product xy is constant
• Graph is a hyperbola (not a line)
• xy = k for all points

Example 1: Is It Inverse Variation?

Given: (2, 12), (3, 8), (4, 6)

Check products:

• 2 × 12 = 24
• 3 × 8 = 24
• 4 × 6 = 24

All equal to 24! → Yes, inverse variation with k = 24

Equation: y = 24/x or xy = 24

Example 2: Not Inverse Variation

Given: (1, 10), (2, 8), (3, 6)

Check products:

• 1 × 10 = 10
• 2 × 8 = 16
• 3 × 6 = 18

Products not constant → Not inverse variation

Finding k for Inverse Variation

Given one pair of values:

k = xy

Example: Find k and Equation

y varies inversely with x. When x = 5, y = 8.

Find k:

k = xy = 5(8) = 40

Equation: y = 40/x or xy = 40

Find y when x = 10:

y = 40/10 = 4

Solving Inverse Variation Problems

Example: Speed and Time

Time varies inversely with speed (for fixed distance).

Trip takes 4 hours at 60 mph. How long at 80 mph?

Find k:

• k = 60(4) = 240

Find time at 80 mph:

• 80t = 240
• t = 3 hours

Answer: 3 hours

Example: Workers and Days

Number of days varies inversely with number of workers.

5 workers finish in 12 days. How long for 15 workers?

Find k:

• k = 5(12) = 60

Find days:

• 15d = 60
• d = 4 days

Answer: 4 days

Direct vs. Inverse Variation

Joint Variation

Joint variation: One variable varies directly with two or more variables

Formula: z = kxy

Example: "z varies jointly with x and y"

Example: Volume

Volume of cylinder: V = πr²h

• Varies jointly with r² and h
• k = π

Combined Variation

One variable varies directly with one and inversely with another

Formula: y = kx/z

Example: "y varies directly with x and inversely with z"

Example: Speed, Distance, Time

Speed = Distance/Time

s = d/t

• Varies directly with distance
• Varies inversely with time

Real-World Applications

Direct variation:

• Cost and quantity (unit price constant)
• Distance and time (constant speed)
• Wages and hours (constant rate)

Inverse variation:

• Speed and travel time (fixed distance)
• Workers and completion time
• Pressure and volume (Boyle's Law)

Practice