Direct and Inverse Variation

What is Direct Variation?

Direct variation: As one variable increases, the other increases proportionally.

Formula: y = kx

Where:

• k = constant of variation (constant of proportionality)
• k ≠ 0
• Relationship is linear through origin

Key phrase: "y varies directly as x"

Example 1: Distance varies directly with time (at constant speed)

If speed = 50 mph (this is k): d = 50t

Travel 2 hours: d = 50(2) = 100 miles Travel 5 hours: d = 50(5) = 250 miles

Example 2: Cost varies directly with pounds

If apples cost 2 dollars per pound (k = 2): C = 2p

Buy 3 pounds: C = 2(3) = 6 dollars Buy 7 pounds: C = 2(7) = 14 dollars

Characteristics of Direct Variation

1. Passes through origin (0, 0)

When x = 0, y = k(0) = 0

2. Constant ratio y/x = k

Example: y = 3x

x = 1 → y = 3 → ratio: 3/1 = 3 x = 2 → y = 6 → ratio: 6/2 = 3 x = 4 → y = 12 → ratio: 12/4 = 3

The ratio is always k!

3. Graph is a straight line through origin

Slope = k

4. As x doubles, y doubles

If y = 5x: x = 2 → y = 10 x = 4 (doubled) → y = 20 (doubled)

Finding the Constant k

Given one pair of values (x, y)

Step 1: Use y = kx Step 2: Substitute known values Step 3: Solve for k Step 4: Write equation

Example 1: y varies directly as x. When x = 4, y = 12

12 = k(4) k = 3

Equation: y = 3x

Example 2: y varies directly as x. When x = 5, y = 30

30 = k(5) k = 6

Equation: y = 6x

Example 3: Distance d varies directly with time t. When t = 3, d = 165

165 = k(3) k = 55

Equation: d = 55t (speed is 55 mph)

Using Direct Variation to Solve Problems

Example 1: y varies directly as x. If y = 15 when x = 3, find y when x = 7

Step 1: Find k 15 = k(3) k = 5

Step 2: Use k to find new y y = 5(7) = 35

Example 2: The cost C of bananas varies directly with weight w. If 2 pounds cost 3 dollars, find the cost of 5 pounds

Find k: 3 = k(2) k = 1.5

Find new cost: C = 1.5(5) = 7.50 dollars

Example 3: Earnings E vary directly with hours h worked. If you earn 120 dollars for 8 hours, how much for 12 hours?

Find k: 120 = k(8) k = 15 (hourly wage)

For 12 hours: E = 15(12) = 180 dollars

What is Inverse Variation?

Inverse variation: As one variable increases, the other decreases proportionally.

Formula: y = k/x or xy = k

Where:

• k = constant of variation
• k ≠ 0
• Relationship is hyperbola (curve)

Key phrase: "y varies inversely as x"

Example 1: Time varies inversely with speed

To travel 120 miles (k = 120): t = 120/r

At 30 mph: t = 120/30 = 4 hours At 60 mph: t = 120/60 = 2 hours Faster speed → less time!

Example 2: Workers and time to complete job

To paint house requiring 24 worker-hours (k = 24): t = 24/w

1 worker: t = 24 hours 2 workers: t = 12 hours 4 workers: t = 6 hours

More workers → less time!

Characteristics of Inverse Variation

1. Product xy is constant

xy = k (always the same)

2. Does NOT pass through origin

Cannot have x = 0 (division by zero!)

3. Graph is a hyperbola

Two curves in opposite quadrants

4. As x increases, y decreases

Example: y = 12/x

x = 1 → y = 12 x = 2 → y = 6 (x doubled, y halved) x = 3 → y = 4 x = 4 → y = 3 (x doubled again, y halved again)

5. Asymptotes at x = 0 and y = 0

Graph approaches but never touches axes

Finding k for Inverse Variation

Given one pair (x, y)

Use: k = xy

Example 1: y varies inversely as x. When x = 3, y = 8

k = 3 × 8 = 24

Equation: y = 24/x or xy = 24

Example 2: y varies inversely as x. When x = 6, y = 2

k = 6 × 2 = 12

Equation: y = 12/x

Example 3: Time t varies inversely with speed r. When r = 40, t = 5

k = 40 × 5 = 200

Equation: t = 200/r (200-mile trip)

Using Inverse Variation to Solve Problems

Example 1: y varies inversely as x. If y = 6 when x = 4, find y when x = 8

Find k: k = 4 × 6 = 24

Find new y: y = 24/8 = 3

Example 2: Time to complete a job varies inversely with workers. 3 workers take 8 hours. How long for 6 workers?

Find k: k = 3 × 8 = 24 worker-hours

Find new time: t = 24/6 = 4 hours

Example 3: Pressure and volume inversely related (Boyle's Law). When volume is 20 liters, pressure is 5 atmospheres. Find pressure at 10 liters.

Find k: k = 20 × 5 = 100

Find new pressure: P = 100/10 = 10 atmospheres

Direct vs. Inverse Variation Comparison

Direct Variation (y = kx):

• Both increase together
• Both decrease together
• Ratio y/x is constant
• Graph: line through origin
• Example: More hours → more pay

Inverse Variation (y = k/x):

• One increases, other decreases
• Product xy is constant
• Graph: hyperbola
• Example: More speed → less time

How to identify:

"Varies directly" or "proportional to" → y = kx

"Varies inversely" or "inversely proportional" → y = k/x

Joint Variation

One variable varies directly as the product of two or more variables

Formula: z = kxy

Example: Area of rectangle varies jointly as length and width

A = lw (here k = 1)

If l = 5 and w = 3: A = 15 If l = 10 and w = 3: A = 30 (length doubled, area doubled)

Example 2: Volume of cylinder varies jointly as radius squared and height

V = πr²h (k = π)

Combined Variation

Mix of direct and inverse variation

Formula: z = kx/y

z varies directly as x and inversely as y

Example: Speed varies directly as distance and inversely as time

s = d/t (k = 1)

More distance → more speed (direct) More time → less speed (inverse)

Example 2: z = kxy/w

z varies jointly as x and y, and inversely as w

Given: When x = 2, y = 3, w = 6, z = 4

Find k: 4 = k(2)(3)/6 4 = k k = 4

Equation: z = 4xy/w

Real-World Applications of Direct Variation

1. Wages: Pay varies directly with hours worked E = rh (r = hourly rate)

2. Distance: Distance varies directly with time (constant speed) d = rt

3. Spring stretch: Force varies directly with distance stretched (Hooke's Law) F = kx

4. Currency exchange: Dollars vary directly with euros d = ke

5. Perimeter and side: Perimeter of square varies directly with side P = 4s (k = 4)

6. Cost and quantity: Total cost varies directly with items purchased C = px (p = price per item)

Real-World Applications of Inverse Variation

1. Travel time: Time varies inversely with speed t = d/r (d = distance is constant)

2. Work completion: Time varies inversely with workers t = k/w (k = total worker-hours)

3. Gas law (Boyle's): Pressure varies inversely with volume (constant temperature) P = k/V

4. Brightness: Light intensity varies inversely with distance squared I = k/d²

5. Gear ratios: Speed varies inversely with number of teeth s = k/n

6. Seesaw balance: Weight varies inversely with distance from fulcrum w₁d₁ = w₂d₂

Recognizing Variation from Tables

Direct Variation Test: Check if y/x is constant

x | y | y/x 1 | 3 | 3 2 | 6 | 3 3 | 9 | 3 4 | 12 | 3

Constant ratio → Direct variation! (y = 3x)

Inverse Variation Test: Check if xy is constant

x | y | xy 1 | 12 | 12 2 | 6 | 12 3 | 4 | 12 4 | 3 | 12

Constant product → Inverse variation! (y = 12/x)

Neither:

x | y | y/x | xy 1 | 5 | 5 | 5 2 | 8 | 4 | 16 3 | 11 | 3.67 | 33

Neither constant → Not direct or inverse variation

Graphing Variation

Direct Variation (y = kx):

• Straight line
• Passes through (0, 0)
• Slope = k
• If k > 0: line rises left to right
• If k < 0: line falls left to right

Inverse Variation (y = k/x):

• Hyperbola (two curves)
• If k > 0: curves in quadrants I and III
• If k < 0: curves in quadrants II and IV
• Never touches axes (asymptotes)
• Symmetric about origin

Common Mistakes to Avoid

1. Confusing direct and inverse Direct: y = kx (multiply) Inverse: y = k/x (divide)

2. Forgetting to find k first Always find k before solving for new values!

3. Using wrong formula Read carefully: "directly" vs "inversely"

4. Arithmetic errors with fractions Be careful with division in inverse variation

5. Thinking direct variation has y-intercept Must pass through origin! (0, 0)

6. Not checking units Ensure units make sense in context

Problem-Solving Strategy

For Direct Variation:

1. Write y = kx
2. Substitute known pair to find k
3. Write equation with k value
4. Use equation to find unknown

For Inverse Variation:

1. Write y = k/x or xy = k
2. Substitute known pair to find k
3. Write equation with k value
4. Use equation to find unknown

Identifying type:

• "Varies directly" or "proportional" → y = kx
• "Varies inversely" or "inversely proportional" → y = k/x
• Check if y/x constant (direct) or xy constant (inverse)

Quick Reference

Direct Variation:

• Formula: y = kx
• Graph: line through origin
• Test: y/x = k (constant ratio)
• As x ↑, y ↑

Inverse Variation:

• Formula: y = k/x or xy = k
• Graph: hyperbola
• Test: xy = k (constant product)
• As x ↑, y ↓

Finding k:

• Direct: k = y/x
• Inverse: k = xy

Joint: z = kxy (varies jointly as x and y)

Combined: z = kx/y (direct in x, inverse in y)

Practice Tips

• Always identify variation type first
• Find k before finding other values
• Check if answer makes sense (direct: both increase, inverse: opposite)
• Practice with real-world contexts
• Graph to visualize relationships
• Use tables to test for variation type
• Remember formulas: y = kx vs y = k/x
• Master fraction operations for inverse variation
• Verify by checking if ratio or product is constant
• Apply to physics, chemistry, economics problems
• Don't confuse with linear equations (direct variation is special case)
• Understand the meaning of k in each context
• Practice converting word problems to equations

Understanding variation helps you model real-world relationships mathematically. These concepts appear throughout science, economics, and engineering!